I taught therapeutic riding for a number of years so its a topic near and dear to my heart
Before being accepted into a reputable program, your doctor will need to sign off and usually your physio. Sometimes the programs physio will also do an assessment on your first visit to help choose equipment suitable for your needs. The riding instructor will also assess you needs for a mount.
All this assessment is very important for the therapy... if you are prone to dizziness, riding may not be for you, depending on the type and frequency of dizziness. If you are unbalanced, a particular type of horse will be (attempted) to be chosen for you. One that has a steady (slow) walk and doesn't transfer too much movement through its back into your pelvis and trunk. The horse should be wider through its back for a wider base of stability.
Lots of factors will be taken into consideration, those are just a few examples above, even just trying different tack can make a huge difference to someone that's unbalanced.
So there isn't necessarily lots of "bobbing about" depending on the horse, some have more active walks and will transfer more movement through to their riders. These mounts are excellent choices for some autistic kids that really need the extra stimulation.
Please make sure if you're going to give therapeutic riding a try that you you choose a reputable program that is a member of Path
http://www.pathintl.org/ or CanTRA
http://www.cantra.ca/ and not just some random person, with a random stable and some random horses.
Always wear a *properly fitted* helmet. Mine saved my life.
Starr