General facts:
1. Knights actually had more than one horse, unless the knight is extremely poor, which would be strange.
2. Tournament horses and riding horses are two completely different horses.
3. Contrary to modern sentimentality, medieval knights did not keep 'special horses'. While the lord would of course have the best horse, for long distances, they would have changed horses at public stables because horses can't gallop for too long. So a knight on his way to London on an urgent trip would have exchanged 'Josephine' for 'Coltmere' at Northampton and he will likely never see Josephine again because someone else would've exchanged their horses for 'Josephine'. That being said, if speed is not an issue, they did use the same horse for the entire journey, with plenty of rest in between. There are also cases where a very wealthy nobleman would bring spare riding horse, so that they can switch to that horse when the one they use is tired.
4. For long distance riding, you would ride a shaggy, kind of gentle horse, that allows you to ride for a long time comfortably as it canters or trots at a sustainable pace.
5. For tournaments, you would ride a big, tough and strong horse. But these horses are very hard on your butt. You definitely need a padded saddle with them.
6. For races, you generally use small but fast horses which can't take much of a load but is made for speed.
7. For horseback knight battles in the late medieval era, you'd preferably want big, tough horse capable of taking on at least 100kg of load without being encumbered. In some cases such as the heavy cataphracts, this could go as high as 150kg. This is because knights of this era typically wears full plate armour, with a thick heater shield, a long sword and sometimes a lance as well. Not to mention the horse too wore heavy armor, either chain or plate.
8. Destriers can't charge at full gallop for more than 2-3 minutes. They have to rest after a full charge. So you will never charge, fall back, charge, fall back and rinse and repeat like in Total War games. They can charge a short distance after the first charge, but their performance significantly drops after charging for 2-3 minutes. Once the charge is broken, fall back and stay out of the battle for at least half an hour.
9. As stated above, a knight would not have only one horse. A knight would be accompanied by his warhorse, his riding horse, is draft animals and his pack mules.
10. A horse had more value as pack animal, so men at arms generally would not fight on horseback. They would dismount and then fight on foot. Because having to carry those heavy armour, supplies and weapons on their own would be very, very tiring.