What students learn when they first start out
According to Pettra Engeländer of the Independent European Horseback Archery School, many riders need to work on their own balance before trying to shoot arrows from the back of a horse. So the basic exercises for mounted archery don't necessarily involve horses. Martial arts from various traditions are included in the training to teach students to "feel and act instead of think and do what somebody says" as Pettra puts it.
The good news is that students who do the work on the ground can expect to be shooting arrows and hitting targets from the back of a horse in only two days.