It depends on the center, what programs/populations they serve, etc.
Professional Association of Therapeutic Horsemanship (PATH) Internationally, formerly NARHA, has standard of practice for horse use. You should be able to find a lot of information from their website regarding horse usage. Sites only have to follow these rules if they are PATH accredited, however, there are sites out there that are not but that does not mean they follow these good horsemanship rules as well.
What the horses gets used for and the “schooling” they receive is very center dependent. If it is primarily therapeutic riding, now called adaptive riding, it will depend if riders require leader/sidewalkers or if riders are working on independent riding skills (trot/canter/basic jumping or dressage) or working towards paradressage goals. Some sites even have adaptive driving and vaulting opportunities. Lessons are mostly groups but could be privates, so more like your typical riding lessons but horses need to adjust to physical, behavior, and learning needs of the riders, so in general more taxing. I thought the rule was no more than 3 hours (maybe 4) with required breaks in between.
Many centers also offer hippotherapy as a treatment tool, which is therapy using the movement of the horse, provided by an OT, PT, or SLP. In these cases, the horses mainly walk for sessions that last 30, 45, or 60 minutes (again site dependent). Clients perform a variety of position changes and functional activities. Some centers even long line the horses for better quality movement while the client is mounted.
A good quality center will have the horses schooled/ridden but either an equine manager on site, trainer, or very qualified volunteer with a good riding resume (though it is hard to find these people). Depending on the area/riding style, they will be worked this way. If not possible, horses will be hacked and or ridden in volunteer type lessons/by less experience riders without disabilities to keep things different for the horse. Again each center is different, some places won’t school their horses, which I think is a necessity for happy horses and keeping the horses fit for their work load for their riders and/or producing quality movement for therapy clients.
Hope this helps. Feel free to ask more questions if you like.