In Confederacy soldiers had to provide their own horses but were compensated if their mount was killed in battle. The problem was they had to be killed in battle to receive compensation and more often than not horses died of exhaustion instead of actual battle. This meant that soldiers wouldn't get compensated and then would have to transfer to the infantry or artillery if they couldn't afford to buy a new mount. If they did get compensated they still had to return home to their community to purchase a new horse. This meant that they were away from their unit for some period of time until they could return with their new horse.

In the north cavalry troopers were provided with their own horse and gear. I'm not sure how compensation and stuff worked for them though I would assume that if they lost a mount they would just be provided with a new one.

While a perfectly fine question, more of our Civil War experts are sticking around in this AMA thread today, so I would suggest you repost it here!