I bought a horse at the stable and grew attached to it, but I want to get a new horse from a different stable. What will happen to my old horse?
Once you ride your new horse, your old one will walk to the stable it was bought from, but it can still be attacked by enemies. Just leave your new horse at the stable you purchased it at and fast travel to the city where you bought the old one. Then just fast travel back to the stable from which you purchased the new one. You can switch them out at anytime by following this process.