Once you’ve already fallen off, here’s what you do:
Calmly walk up to the pony without holding a grudge and catch it.
Calmly lead him back to the mounting block or a place where you can get back on. Do not act pissed or get after him as it will only scare him and that sounds like it is what caused the second episode with the pony. Either your daughter was getting pissy with him or someone who owned him before taught him he would get his @ss beat.
Get back on.
Carry on.
To school the pony, do exactly what you were doing before and THAT is the place where you school the behavior.
I walked this talk two days ago. A four year old got playful in a lead change going away from a jump and sent me absolutely unceremoniously sailing. He did a surprised little courtesy circle, and trotted right back to me. I took the reins, quietly led him back to the mounting block, hopped on, gave him his customary pat for standing well at the block, promptly picked up a nice little canter, and headed immediately back to the same jump, jumped it, and without being rough or holding a grudge halted on the back side, softly backed two steps, patted him for halting and backing well, and then did the whole thing a couple more times before putting him away.
The part where I trained him to try to contain his youthful exuberance a little more was where we quietly halted and backed after the jump. That was where I said, “BooBoo, you need to stay focused a little better after the jumps, ok?” THAT was the training moment.
Do not set up the appurtenant training moment for disaster by getting after a horse on the ground before you get to the training moment. Get to the real training moment as quietly, smoothly and undramatically as you can so the horse has half a (calm, quiet) brain to LISTEN to the training moment when it occurs.
They are just horses.
Getting their @ss beat from the ground doesn’t teach them anything.
Getting back to WHERE THE PROBLEM OCCURRED and quietly showing them what you DO want them to do does.
And when he does it well again, pat him and put him back in his house.