I went to a big equine school that would get dozens of donations every year. We kept (and still do) over 100 horses on the property. Bigger schools with better programs have stricter qualifications for the horses they bring in. Schools without a major or with a smaller program will be more likely to take and keep horses that aren’t very very useful (as in jumping 3’6"+, high level dressage horses, or absolute packers). Most schools take very good care of their horses, but know the program. My school had a fantastic head of the program when I was there who always put care of the horses first. Now they have an equally fantastic barn manager and a vet on staff. Some schools rely solely on the students to take care of the horses (and we all know how that can be a problem…). Although with bigger programs (like ours), you have to recognize that the horses are teaching tools - they need to be able to handle multiple lessons a week and sometimes horse shows on the weekend. Some horses just can not handle this program. Many schools (mine included) will make the effort to contact you when your horse is no longer useful IF you specifically as them to & put it in writing in the form of a letter they can put in their file. If you don’t do this, it is to difficult to track people down after an extended period of time when they have so many horses coming in and out. Students can also ask to take a horse upon retirement by the same process of writing a letter. That’s how I got my horse who is happily living out his days in a pasture about an hour from me & on the other side of the country from where I went to school 
Of the horses I knew about our school getting rid of, most went to former students or friends of staff and students. Some went to other schools. One went to an equine science department. A few were sold (yes they CAN turn around and sell the horses that are donated). And, yes, some did go to auction. Those horses however were unwanted by their former owners, unusable in the program, and unplaceable elsewhere. You can’t expect these schools to bend over backwards to give every horse a life of rainbows and sunshine when they can no longer use them. Many of these horses are ‘dumped’ at these schools, and they aren’t allowed to ‘dump’ them afterwards?? If you are unable to sell your horse or use them anymore when you are done with them and donate them, you can’t really blame a school for having the same trouble doing something with them when they are done with them.
If you are donating your horse because they have an injury that keeps them from passing a vet, but they are still very useful animals (the majority of the donations my school received) - then you can be safe to assume that your horse will have a great life in a university program and someone will probably fall in love with them and take them after retirement. On the opposite side - if your horse is lame, sour, a dirty stopper, a rearer, a spinner, a bucker, a problem child, or has any other various serious issues - a school will have just as hard a time using them and placing them as you did. These horses, unfortunately, will be the ones to bounce around from place to place to not so nice place. If you can’t afford to give your horse the retired life you want them to have, you can’t really expect someone else to do that for you.
