Ann Hyland wrote two books on The Warhorse with different subtitles re the time periods covered: the first ancient and medieval, the second Renaissance and modern. She also wrote Equus on horses and mules in the Roman realms.
First, wash out any images of knights on Shires and other big draft horses. Draft horses are for draft, are weak in the back for carrying, have gallumphing gaits, and only developed in the later 1700s, but mostly in the 1800s. Shires get their height from Thoroughbred blood. (Silver, Horses of the World)
Thoroughbreds were developed from the 1600s forward as cavalry remounts (Osmer, On Horses), and one of the qualities desired was speed. So they developed as racehorses. They are the fastest breed.
Now, you can erase all this modern stuff because medievals didn't know about it. While Svinhufnud did translate an Anglo-Saxon horse care text, we can see from that the medievals were still largely basing choices in riding horses on Xenophon, On Cavalry, c.350 BC. (You can get a translation at Project Gutenberg or the Perseus Project.) What makes a good horse for the job had not changed.
A good-sized warhorse was 15hh, and a hand is 4", measured to the withers, the pointy bit behind the neck. Hyland bases a lot of her sizing on horseshoe sizes, and surviving saddletrees. Smaller was common. The look of the horse was like a heavy hunter. Roman noses were the norm. Look at statuary. The Frisians of today preserve much of the look, and so were used in the movie Ladyhawke. A Frisian/Saddlebred cross is possibly closer, and a completely droolworthy beast.
Warhorses were called destriers/dextriarii because they went on a dexter lead (our left lead), according to S. A. Bolich. This put the support in the right place for crashing your lance into someone else. On the right lead, the horse is far more likely to fall back. So they took special training.
It's controversial whether they had haute ecole training (like the Lippizzaners in Vienna), if this would be any use in battle, but a horse fighting footsoldiers under saddle is reported as far back as the Graeco-Persian Wars. The Athenians had a special bounty out on Mardonios's warhorse because it was such a terror. You can also see haute ecole poses in the Elgin Marbles. Bolich says it could work, and it would seem to be required to take on a phalanx.
As to travel: depends on breed, condition, supplies, and roads. Not to mention climate, weather, and necessity. Some Mongol horses probably travelled from Mongolia to Europe. I would need my notes at home to quote weights of armour for the destrier.