Hey, I've been learning how to spar, and currently I've been focusing on mobility. I have the horse stance down, and I'm decent at running, but I'm having a little trouble stopping out of a run and going back to horse stance. I pretty much always end up falling over. Can anybody help?

First thing I noticed is that you watched mocucha's video on how to spar. That isn't a very good tutorial knowing that the horse stance is now frowned upon.
I'd heavily suggest you try and develop your own unique style. Me for example, I have a jumping and kicking style as opposed to being on the ground punching. That's what I'd say you try first. If you do want to have that punching style, then you need to master the horse stance aswell as other techniques on how to stay balanced when hitting your oponent. I hope this helped.

Put your weight behind you when you stop so that the momentum trnasfers to the floor rather than to keep pulling you along, almost like youre skidding to a stop on ice skates.

you have to have your feet in front of you and lean slightly backwards if youre trying to stop you forward momentum, your generally not just going to stop out of a run like that, the main thing that will happen is youre going to want to just try and slow yourself down enough to shift the direction of you momentum and then stop. atleast thats what ive gotten from my experiences so far

also dont listen to people telling you certain things are frowned upon, just do whatever looks nice tbh