If you're arrested while riding a horse, what do they do with the horse?

Obviously there's gonna be different protocol for different scenarios. I grew up in very small town America, our "town" had more bars than cops, more churches than bars, and no grocery store. People rode their horses around town frequently, even dropping their kids off at school via pony. What would be the protocol if someone were arrested while riding a horse? Does it matter how far they are from home? Where does the horse go? Who's responsible for it? Can it be impounded or auctioned off like a car? Are the next of kin called?