The answer can be pieced together quite well from the Wikipedia page, in my opinion, so I did that. It changes somewhat from place to place, here is the British development.
Essentially, when Chess arrives in Europe at around AD 1000, it has the following differences from now:
- Pawn moves one square forwards only, and can be promoted to queen only.
- The Bishop is called "elephant" and moves two squares diagonally, but jump over a piece in-between.
- The Queen is called an "Adviser" and can move only one square diagonally.
- The Rook is called "Chariot".
- Stalemate was a type of inferior victory for the person initiating it.
- No castling.
By the 12th century the pieces had gotten their modern names, and the rule that the Pawn could move two steps as it's first move had been added. There are some rules (that seem to change over time) for protecting the king by letting him jump towards a corner.
So your first checkpoint at 1450 means the major differences from the modern game is that the Queen and Bishop have very limited movement, and when playing for money, initiating a stalemate is a win that gives you only half the winnings. You can also only promote Pawns to Queens, and there is no castling, but some sort of precursor.
The "en passant capture" also first showed up during the 15th century (or possibly late 14th century), but it seems to have been controversial and not universally accepted until the 19th century. (ref) So if you included that rule or not was likely dependent on location.
By 1530 the Queen and Bishop had acquired their modern powerful moves. There would have been no significant changes by 1610, but the view of stalemate would have changed, and the one initiating the stalemate would now lose.
Changes since 1610 is that stalemate is now viewed as a draw and the modern rules for castling and promotion arrived.
This means that the game in 1450 would have played very differently. The weak Queen and Bishop means the game would have played much longer, and slower, and it would be more common with slow games played over several days. However, the 1575 and 1610 games would be very much like today’s games, and most of the modern strategies would apply then as well.
Information condensed from: