Hey! I was volunteering at a therapeutic riding stable and some of the games we did on horse back included...
-we had a post box and the instructor made notes and put them in there, the kid would ride up open it and choose a slip and there would be instructions on the slip like "sit facing the horses tail for one lap of the arena" or "trot a 20 metre circle" or "sing "insert song here" and do the hand actions while walking a lap of the arena" etc
- we had a mini basket ball hoop and a ball with letters. We through the ball to the kid and whatever letter was on top they have to think of some words with that letter or just read the letter out, depending on their age and ability I guess, then try to throw the ball into the hoop. We also have a ball with animals on it and they could identify the animal and the sound in makes then throw!
- we had lots of puzzles, where we would place the puzzle pieces all over the arena, like in buckets hanging on posts or the side of the arena etc and they had to ride to the pieces then take them to the puzzle board and make the puzzle. We even had some puzzles that made noises when you put the piece in, they were popular with the younger kids.
- we had the kids sit in all sort so positions, like facing sideways, facing the tail, up in all fours etc
- then lots of ball games where they caught and throw while the pony was walking or throwing the ball at an object to knock it over. We even had a hockey stick and balls on cones that the kid would reach down and hit the ball off the cone.
- we would quiz the kids of parts of the pony. Like where is the mane, ears, tail etc and how many hooves etc
That's all I can thin of for now but each instructor had different things for each kid.