Often, these horses can go into a new career. If your horse still can work, then you might be able to sell him to a rider who needs to school with an experienced mount. A riding camp might be another outlet, but might not serve your horse well in the long run. Some of these young would-be cowboys and cowgirls might needlessly be thumping on the horse's sides to make him gallop, pulling hard on the bit to steer or stop him, and bouncing all over his back. If the horse ends up resold at auction, chances are great his remaining days will consist of being crowded on a trailer with strange horses bound for slaughter.
Similar problems can occur with horses sold to riding schools.
The riding school is going to look at it as a liability because it is in the business of providing horses for people to take lessons on. Very few riding schools have the ability to keep amassing old-timers that can't be used. Those horses would probably be sold to the highest bidders, which would probably be, for an unsound horse, the meat industry. Auctions, while offering another outlet for selling a horse, can end up being the equivalent of death row for your horse, if the selling price is within the range of slaughterhouse reps who buy cheap horses for their value as meat products. But you might not sell the horse.