On occasion, when there are riderless horses on the battlefield, either because allied riders have fallen or enemy horsemen killed, you can order your troops to mount up. (Command key 4 or 5)
Now, they aren't always able to follow the mount up command. Sometimes they just run at the horse and against it like they're on a treadmill. I've never investigated why they don't jump on, it may be that particular trooper has no riding skill but is still trying to obey the order.
I don't recall 100% but they won't keep the horse after battle or I think after the next battlescreen where you keep the option to fight or flight. And as you know, any enemy horse you're riding on, won't necessarily appear on the potential loot inventory either.