“ . . . Would there be anything different a gun crew would be doing aboard a ship . . .”
Yes, of course, the gun crews did not sit around to their guns all day on the chance the enemy may come over the horizon. Easily 95% of the time at sea was spent under plain sail going from point A to point B or, perhaps, maintaining one’s position in a blockade squadron. This required sailors to man the yards and perform maintenance 24/7. The crew was divided into watches with, presuming no monster blows or other non-combat hazards, responsibility for generally performed ship handling and maintenance duties during ones’ watch. This applied to officers as well.
If not on watch, one could be below relaxing, even sleeping. If divided into two watches that meant 4 hours on and 4 hours off (except for the dog watches which were 2 hours in length and served to ensure a watch was not always serving the same hours).
See https://www.rmg.co.uk/stories/topics/what-are-watches-on-board-ship for watches.
Some ships, with sufficient manning were divided into three watches, certainly a bonus in down time, but this was a very unusual practice. In truth, the two watch method predominated. Under smooth sailing conditions the middle watch, 2400 to 0400, was probably the closest to down time on deck. The watch was there and available, but not scampering about the rigging (except to make changes as called, but they might not touch a backstay for an entire watch if conditions are right) or holystoning the deck. In the daylight hours there was always plenty to do. And when going aloft, a sailor did not, except in an extreme emergency, did not just pick a mast and go, he was assigned to a specific mast and worked in a team or teams on that mast, thus there were foretop men, main mast men, and so on.
One might note that lacking a full complement (not at all unusual) gun crews might have to fight both sides of the ship as the situation calls. They would also comprise most of those making up boarding parties when called for boarders away.
Ideally, one would want one’s general crew to be able seamen, who could hand, reef, steer, and handle the ships guns . . . all nice round pegs in round holes.
Just as in modern navies, one does not man the guns except at action stations; the rest of the time is spent on routine maintenance tasks on watch and trying to relax or sleep when off watch. One of the most precious commodities on a ship, sailing or powered is sleep.
MUCH MUCH LATER 6-12-21
Took me a while to put this together, my source not lending itself to cutting and pasting, but, in response to your clarifying comment of 7 June . . .
The answer is “it depends.” Presuming the ship has an optimal number of crewmen, how many are assigned to a gun and what they do depends entirely on the size of the gun, from a, say, 64 pounder, requiring more muscle power, to a swivel gun which needs at most two men and can probably get by with one. Also presume you are addressing the sailors on the Constitution operating what looks to be a long 24 pounder. Both Schwern and Pieter Geerkens correctly point out that the carronade as opposed to a standard gun require smaller crews due to size and weight and the improved recoil dampening and traversing capability . . . or course at the cost of a much shorter ranged weapon.
Major General (yes, a British Army Major General) Sir Howard Douglas, Bart., KSC CB FRS, writing with the authority of the Admiralty produced A Treatise on Naval Gunnery the second edition of which was published in 1828. The treatise is actually an enjoyable little read though if you’re not familiar some of the technical details, I suppose, could be a little daunting. Lots of interest discussion on gunnery schools, weapons, ballistics, and such and then, he gets to the meat of the matter . . . shipboard gunnery.
https://archive.org/details/in.ernet.dli.2015.109568/page/n1/mode/2up
Douglas specifically mentions a Royal Navy contemporary of the Constitution, HMS Shannon, equipped with long 18’s I believe, of which he has some interesting words. He points out that before 1817 there was no Admiralty established system of gun exercise and his belief that there should have been. Anyway, he writes, first about the problem of calling gun crew members away from their guns for specific purposes if basing such calls on the number of the gun and the number of the individual sailor in his assigned gun position, starting on page 145. This is long, but hopefully you will see where it is going:
“120. It appears to me, that the mode of stationing and
distinguishing the crews of ordnance in the French code, is extremely
simple, and preferable to the practice of designating the people by
numbers throughout. The principle of numbering off the crew of a gun,
as practiced very properly in the land service, has of late been a
good deal followed in the navy, and it forms the basis of the new
system to which I have already alluded. Thoughtfully impressed with
the advantages of such an arrangement for land service, I more than
doubt the expediency of adopting it generally in the naval service.
Artillery soldiers numbered off to a gun, are not liable to be called
away suddenly in the heat of action to the performance of other duties
assigned them also by numbers. But naval gunners are told off not
only to their gun duties, but to other very important callings, some
of which must be designated by numbers, as far at least in two or
three places. When calls, by numbers, are made upon men station by
numbers to their guns, it may fairly be apprehended that some
confusion or hesitation my ensue, particularly if that call should be
made after having sustained any casualty that may disturb that
division of labor, and order of succession to other duties which were
originally provided for in the numerical classification. For various
manœuvres of tacking and wearing, making and shortening sail, trimming
sales, boarding, extinguishing fire, small-arm duties, etc. the men
are told off by numbers, as far as two or three places at least.
Thus, 1st sail trimmers, or 1st and 2d small-arm men, or 1st and 2d
boarders, may be called; but they may have have changed their original
gun number more than once, and all this complication of numbering is
burthening the men’s memories in a manner that may occasion some
hesitation in obeying such calls, and some confusion in the substitute
arrangements at the guns. These inconveniences and objections to
numerical designation were so strongly felt in the service, that in
some of our best men-of-war the practice was abandoned, and the men
stationed, as much as possible, by appellations expressive of their
duties. This is the principle also of the French Manual, in which the
men are called chiefs, chargers, loaders, providers, 1st and 2d
right-men, 1st and 2d left-men, etc.
“In the Shannon the men were designated 1st and 2s captains, loaders,
loader’s mate, 2d loader – these taken from the ablest and most
intelligent men; then tackle-men, powder-men, lantern-men. From the
first class were taken boarders, sail trimmers, and firemen, when
called. From the second class, or mere assistants, were taken
small-arm men, etc.
“For action on both sides, the 2d captain of each gun becoming acting
captain of the opposite piece, and the 2d loader, it’s charger, and so
on. The crew of the guns were told off to starboard and larboard
watch, alternately, so that they might be exercised in complete
complements, in their respective watches, and thus know, and have
confidence in each other, and be accustomed to act together. Then
each gun-crew was subdivided into two classes; and whenever, as at
night or otherwise, it became necessary to prepare the guns for
action, without however disturbing the watch below, the second
division of each gun-crew upon deck cleared and prepared the adjoining
piece belonging to the other watch, who were not consequently turned
out till called for action. In long anxious chases of large vessels
on uncertain appearance, these excellent arrangements must have been
of infinite use, in sparing the men from fatigue, without relaxing
anything of service efficiency.
“Much might be said upon the importance of stationing men to duties
about the guns by appellation, in preference to the arrangement by
numbers, and upon the necessity of making this appropriation
subservient, not only to the talents and fitness required for duties
about the gun, but to the qualities and powers required for other
duties to which the men may be called. All this demands intimate
knowledge of the several qualities of the whole crew – occasional
alterations in proportion as these are discovered, and, consequently,
various modifications in the manual system. This view of the subject
again shows the indispensable necessity of intrusting the instruction
of naval gunners to naval characters only: for the arrangement of
duties about the guns is intimately mixed up with that which best
provides to duties of seamanship and manœuvring, with the least
possible interruption to the service of the ordnance; and these are
altogether formed from data more favourable to the arrangement
appellation than to the system of numbers.”
In case it is not clear, Douglas is approving the common naval practice of naming each position, however succinctly, over the common land artillery designation of crew positions by simple numbers. We can see clearly that aboard USS Constitution the practice of naming the positions was followed:

This a gun roster, for a 24 pounder in fact, which can be found on the US Naval History and Heritage Command web site https://www.history.navy.mil/research/library/online-reading-room/title-list-alphabetically/c/the-constitution-gun-deck.html
The thing to remember is that in a gun crew everyone has a specific job. Handling a gun does not mean just firing it effectively, it also means safely. No one interested in innovation. What each person does, and in what sequence, was a carefully choreographed dance that included both efficiency and safety. While no doubt in different navies there might be slight variations, everyone pretty much did it the same way. Large guns in the 17th century had slightly different carriages, but not so different as to drastically change the steps of the dance. The same is true today everyone has a specific job to be performed at a specific time(s) in a specific way. Of course, each crewman also needs to know what his counterparts are doing and be ready to step up in the event of a casualty. But never snatch the rammer from the first man on the right, the rammer, and say “let me, I’ll do it !” No, no, not your job and means someone else has to your job, with a domino effect. Plays hob with crew efficiency.
So, what would be a gun crew on a ship in the 18th-19th century? Well, adapted from page 149 of A Treatise on Naval Gunnery 2d Ed., published in 1828:

So, for a 24 pounder, it would appear that the gun crew shown in the film is actually somewhat smaller than optimum, though sufficient for an 18 or 12 pounder. A carronade of any size with taut breechings can be easily operated by a crew of four, including the captain of the gun, with less steps required to service the piece.
So, what do these fellows serving a 24 pounder do? Quoting directly from the above mentioned treatise, starting on page 150:
“The captains of guns command, and point. They are placed behind the
breech. The privates, equally divided, are placed on each side, and
the in the following exercise are called right-men and left-men. The
first man on the right spunges and rams-down cartridge. The first man
on the left loads, receiving the cartridge form the powder man, who
fetches it during the action, placing himself behind the man who
loads, holding the cartridge-box carefully covered.
“The last, or rear-man on the right is provided with a small apron,
having a pocket to hold spare flints, and some ole linen for cleaning
the lock.
“On board line-of-battle ships, frigates, and sloops-of-war, two
gunner’s mates in each battery will be provided with bags, containing
a stock and bits, four gimlets, one screw-driver, two spare locks,
spar line for trigger pulls, and some old linen for cleaning the
locks.
“On board brigs and vessels carrying less than ten guns, one gunner
carrying a bag thus provided will be sufficient.
There were thirteen specific commands to be given to serve a gun.
Continuing from the treatise:
“EXERCISE OF THE GUNS
“Observations.
“122. As the guns ought always to be loaded when ships are out of
harbour, the Exercise begins according to this supposition. Upon the
general command to repair to quarters being given, the powder-men
proceed, in conformity with previous instructions, to the gunner’s
store to fetch the powder-horns, tube and cartridge boxed, etc. The
two last men on the fight fetch the locks and every article with which
the ought to be provided. The other men repair to their respective
guns; dispose every implement ready for action; cast loose the guns,
and remove every obstacle that might impede their manœuvre. The train
tackles are hooked to the cruppers of the carriages, and two the
ringbolts in the rear.
“Through the whole of the exercise for instruction, the commanding
officer explains, after each word of command, the number of motions in
which the manœuvre is to be done. The men listen with attention and
do not begin to execute the intended manœuvre until the word action is
given.
“Upon the signal being given that the exercise is to commence, the
greatest silence must be observed. The captain of each gun faces to
the port-holes, the men on the right and left pace their guns dressing
by the first two men, and closing to the ship’s side, so as to feel
slightly the elbows of each other; heads up, eyes fixed on the captain
of the gun, the body upright, arms hanging down, hands open and flat
upon the thighs.
“FIRST COMMAND.
“123. Take out your Tampions, cast loose your Guns.
“The first man on the right takes out the tampion and places it behind
him, close against the ship’s side. The captain of the gun with the
assistance of the men near him casts loose the gun, secures it against
the ships side by hitching to the cascable both tackle falls, which he
then gives to the 2d men right and left to hold; he then takes off the
vent-apron and hands it to the 3d man on the right, who places it
against the ship’s side.
“SECOND COMMAND.
“124. Prime you Guns. (In one motion.)
“The captain of the gun lays hold of the priming wire with his right
hand, pierces the cartridge, and ascertains, by moving his wrist, and
by the length of the wire, whether cartridge be home or not; he then
takes a tube out of the tube box (which he shuts quickly), tears off
the cap of the tube, and introduces into the vent. He then takes the
priming-horn in his right hand, opens the pan of the lock with his
left, fills it with powder, shuts it quickly, and replaces the
priming-horn behind him.
THIRD COMMAND.
“125. Point your Guns. (In three Motions.)
“1st Motion. – The captain of the gun places himself to the right of
the train tackle, his left foot advanced and flat on the deck, the
knee bent, his right leg stretched backwards, his left hand laid on
the base-ring of the gun, and his right hand on the handle of the
pointing wedge or quoin. The 3d man, assisted by the 4th for heavy
guns, takes up the hand-spike and the iron crows, places them upon the
steps of the carriage to raise or lower the breech, as the captain of
the gun may direct, until the gun have a proper degree of elevation.
“2d Motion – The same men then place the hand-spikes and crows under
the cheeks of the carriage, to traverse the gun left or right,
loosening the tackle-falls, and handing them to the men placed nearest
to them, who, with the assistance of those who are not employed in
pointing or traversing, keep the gun hauled taught to the port. The
captain of the gun then cocks the lock, takes hold of the trigger line
in his hand, and falls back quickly, beyond the recoil of the gun. He
points stooping and placing his eye in a line with chace-sight and the
breech-ring.
“3d Motion. – The gun being pointed, the captain gives the command –
To your posts. Upon which the men withdraw the hand-spikes and crows,
and fall-in, holding them upright, resting on the deck, so that the
trucks may not run over them when the gun roils.
“FOURTH COMMAND.
“126. Make ready the lint-stock. (In one Motion.)
“The rear man on the left lay hold of the lower end of the lint-stock
with his right hand, and of the top with his left, places himself on a
line with the hind-axletree, facing the port-hole, stoops, in order to
blow on the match, then holds it within four inches of the base-ring
of the gun, ready to fire the moment the captain of the gun gives the
word, if the lock should fail.
“FIFTH COMMAND.
“127. Fire (In two Motions.)
“1st Motion. – If the pointing of the gun with respect to line be
correct, the captain watches for the favourable time to fire. The
moment it offers, he makes it known by a signal; and pulls with a
jerk, the trigger line. If the gun should not go off, and he think
the direction still good, he will give the command, fire, the moment
the roll of the ship allows it. Upon this, the man sets fire to the
tube with the lint-stock, holding it so as not to be higher than the
vent; he withdraws it quickly, and refixes it in its place the moment
the gun has gone off.
“Upon the signal to fire, the men who have hold of the tackle falls
let them go, taking care that they are not in the direction of the
trucks. The men holding the hand-spikes and crows lay them on the
deck. Every man, except the first right and left, move quickly to the
train tackle, hook it to the crupper, and haul upon it if the gun
should not have run in sufficiently. The first man on the right lays
hold of the iron crow by the claw end, to block the trucks, as soon as
the gun is clear of the port-hole – the same man, together with the
1st on the left, clears the tackle-falls and breaching; and the last
man on the left fastens the train-tackle-fall with a clove-hitch.
“2d motion. – The 3d man right and left, with the assistance of the
4th for heavy guns, lay hold of the hand-spikes and crows, place them
under the breech, and heave it according as they are directed by the
captain of the gun, who adjusts the bed and quoin so as to replace the
gun in a position convenient for being loaded. The other men clear
the side and train-tackle-falls. The hand-spike is put back in its
place, the crow laid across the truck, and every man falls into his
place.
“SIXTH COMMAND.
“128. Stop the vent, spunge your Guns. (In two Motions.)
“1st Motion. – The captain of the gun takes the priming wire in his
right hand, and enters it into the vent to ascertain if it be clear.
He then stops it with the thumb if his left hand until the gun is
loaded. The first man on the right places himself by the muzzle,
stepping over the tackles and breaching, and the second man hands him
the spunge, which he immediately rams down to the bottom of the
cylinder.
“2d Motion. – The 1st man on the right having rammed down the spunge
to the bottom of the bore, twists it round several times; and, as in
using the worm, draws it out still turning it in the same direction;
he then lays it on the muzzle, and strikes it three or four times to
shake off any remnants of the cartridge or foulness.
“The captain of the gun introduces the priming wire into the vent to
ascertain that it is clear, and then stops it; the last man on the
right cleans, in the mean time the lock, half cocks it, and falls back
to his post.
“SEVENTH COMMAND.
“129. Prepare to load. (In one Motion.)
“The 1st man returns the spung to the 2d and receives from him the
rammer, the head of he rests on the head of the carriage, holding the
handle with both hands. With light guns the spunge-stave is turned,
as the spunge and rammer are on the same handle.
“EIGHTH COMMAND.
“130. Load with Cartridge. (In one Motion.)
“The 1st man on the left faces to the left, to receive from the
powder-man the cartridge which must be carefully put in with the
bottom first and the seam downwards. The 2d man takes a wad, hands it
to the 1st, who places it over the cartridge. The 1st man on the
right stretches out his right arm at full length, resting his left
hand on the muzzle of the gun, his body inclined a little forward,
ready to ram down. The moment the provider has delivered his
cartridge, he goes for another, holding the cartridge-box under his
left arm, and his right hand on the lid.
“NINTH COMMAND.
“131. Ram down Cartridge. (In one Motion.)
The 1st man on the right rams down three times, then lets go the
rammer, drawing back his body. The captain of the gun introduces the
priming-wire to the vent, to ascertain if the cartridge be home; if it
is not, he gives the command to ram down a second time. When he finds
all is right, he gave a signal with his hand, upon which the man draws
out the rammer and places the head of it on the fore-part of the
carriage as before. In the mean time, the 2d man on the left takes up
a shot, which he hands to the 1st man, and immediately afterwards a
wad.
“TENTH COMMAND.
“132. Shot and Wad your gun. (One Motion.)
“The 1st man on the left puts the shot into the gun, and, to prevent
it from falling out, places his right hand upon the muzzle; with his
left he takes the wad, which is handed to him by the 2d man, and
places it over the shot. The 1st man on the right then thrusts them
down with the rammer, and, ascertaining by the length of the handle
that they are home, informs the captain of the gun of it; then
stretching his right arm at full length, rests his left hand on the
muzzle of the gun, and inclines his body forward ready to ram down.
The 1st and second men on the left fall back to their posts.
“ELEVENTH COMMAND.
“133. Ram down Shot. (In one Motion.) “The 1st man on the right rams
down twice, draws out the rammer, and delivers it to the 2d man, who
lays it down on the deck. If it is a light gun, he turns the stave.
The 1st and 2d man on the right fall back to their posts.
“TWELFTH COMMAND. “134. Run out your guns. (In two Motions)
“1st Motion. – The 1st man on the right unblocks the trucks by
removing the iron crow, and puts it back in its place; then, with the
1st man on the left, lifts the breeching from getting foul during the
operation. The rear man on the left loosens the clove hitch on the
train-tackle, keeping hold of the fall, to slacken it as the gun is
run out. All other men lay hold of the side tackles.
“2d Motion. – The captain of the gun gives the word heave. The men
all pull together to run the gun to square to the breast of the port;
which being done, the captain takes care to secure it, by taking a
turn of the tackle-falls around the neck of the cascable, giving the
ends to 1he 2d man on each side to hold on. N.B. If the exercise is
to be continued, it will be resumed from the second command, art. 124;
if not, it will be concluded by the following command.
“THIRTEENTH COMMAND.
“135. Put in your Tompions. Secure your Guns. (In two Motions)
“1st Motion. – The 3d man on he right hands the vent-apron to the
captain of the gun who lashes it on the breech, and then loosens the
tackle-falls, and gives them to the two rear-men to hold. He then
places the slack of the breeching, which is held up by the second man
on each side, between the cheeks of the carriage, and orders the men
to haul taught the side tackles, which he then fastens with a clove
hitch round the neck of the cascable.
“2d Motion. – The 1st man on the right puts the tampion into the
muzzle of the gun; the other men coil up the tackle-falls, fasten the
coils neatly to the cheeks of the carriage, and put every implement on
its proper place. The rear man on the left unhooks the train-tackle
and lays it on the gun. The powder man carries back the priming horn,
tube, and cartridge boxes, to the gunner’s store. The rear man on the
right carries back the bag apron, with spare flints, etc. and the
lock, unless special directions be given to leave it fixed.”
Probably far more than anyone wanted to know.
Main armament guns were not so different in design from the 17th century to well into the 19th century. Certainly there were improvements, probably the most important being the firelock mechanism (which, yes, does not appear on either of the two examples shown as both guns predate that feature), but, as noted above in the gun crew instructions, there was always somebody standing by with a lint stock or slow match should the lock fail to ignite the charge. Some of the hardware changed, note the wheels on the gun carriage shown first below versus the wheels on the second. Also note the difference in the train tackle for holding the gun back during the reloading steps (see 1st Motion of 127, above).
Restored gun from the "Alderney Elizabethan Wreck" lost in 1592 (found here http://shadyislepirates.com/?q=tudor_sea_tactics)

And for comparison as restored gun, HMS Endeavor, circa 1768, one of several, mostly found in Australia (see https://www.arc.id.au/Cannon.html)

What is readily obvious is that the general designs are not that far different, form following function/purpose, and operation following form. The conclusion one could draw, as John Dallman notes above "If the 17th-century Dutch ship had muzzle-loaders on 4-wheeled carriages, then the tasks its gun crews needed to perform would be the same as for Constitution." No doubt there were variations based on the gun/carriage design, but the essential point is that there are only so many ways to service a gun of this size and each member of the gun crew, regardless of what they were called, had a specific job . . . gun captain, sponger/rammer, loader, ammunition passer, tackle handlers, powder monkey . . . and had to be in a specific place, at a specific time, to perform a specific duty.