I can imagine that in the post-Event world, quadruped returnees skilled in martial arts would rework their skills to fit their new forms. Please remember that in the episode “Call of the Cutie”, Rainbow Dash attempted to teach Apple Bloom a type of pony martial art in a dojo. What do you think are the best martial arts for quadruped creatures? Imagine all the fun to be had in martial arts tournaments between several colonies!

Well... considering that they are very anthropomorphic, but still quadruped, it would be a mix between many. Some kicks and punches could work, but punches would be something that they would use sparingly, because that would make them more vulnerable, and many wrestling (not that shit from TV) moves could be translated.
Kung Fu has low kicks that would work, and part of Judo is about subduing an enemy who is not standing upright.
There is a riding style that has (never used in practice) kicks you can teach to a horse.

Realistically, horses fighting is kind of comical. They are all like „Eeeeeee”
and flailing or kicking or biting at each other.

Having done a few years of self-defense-oriented martial arts training, I feel I can say that a great deal of it is actually pretty generic: Learning how to strike without hurting yourself, where to strike so you don't break when you hit, how to hit a hard target vs a soft one, etc.

A well-trained sensei would likely be pretty awesome even if none of the specific moves they learned were still possible. I love the open palm strikes I learned from Kung-fu, for example, but there's no palm anymore. Given how hooves are shaped I don't think half of those strikes (like the palm heel strike) would work too well, but knife-hand and ridge-hand techniques could work... Even better since now you're using a hard edge of a hoof instead of fragile knuckles, though instead of worrying about broken fingers you would have to worry about chipping a hoof or messing up an ankle.

I think all of the kicks would likely work still, given how flexible pony hindquarters can be, and a buck is rather similar to a back kick in concept, just you have more legs to secure yourself to the ground and use to gain power... Hmm...!

It's always hard to choose one art over another, really. They all have benefits and drawbacks to their core philosophies, and those philosophies won't change with biology even if their move sets do.

Ever since PAP I have been thinking about the martial arts of unarmed combat for ponies. I have some ideas on it based on other stories, There is one fix that comes to mind where it has been done in a limited sense without developing your magic to an extreme. The Artist would eventually have to throw out certain moves that don’t pertain to a horse, but can introduce other techniques based on the new physiology. Just think of the fun that could be had when a transformed dragon is a master of monkey style.

I love this... and we know that it wasn't just Dash, either. Rarity has thrown some around as well...

As to PAP, having taken a number of martial arts over the years, even if I don't actively practice anymore, I'd think Judo and Hapkido would translate fairly well. Much of it is about movement more than striking, shifting weight and momentum to offset your opponent's actions. Likewise, many of the throws, locks, and 'attacks' are little more than moving out of the way and taking advantage of your opponent overextending.
Nothing says "I've got no thumbs" like sidestepping and rolling your attacker over your shoulder, dislocating their limbs and catapulting them into nearby foliage with earth-pony strength.

Likewise, many of the sweeps would be completely viable... we've seen Applejack (and, indeed, most of the cast) capable of bucking with a sharp pivot. All that would be required for a proper sweep would be to avoid the kick and slide the (admittedly harder, stronger, and more dangerous) hooves over the ground before popping back to a normal stance/standing posture.

For Pegasus, many of the aerial movements like kip ups, back flips (which ARE a dangerous attack btw), and spin/high kicks would be excellent.

Capoeira might even be viable for ponies since it primarily emphasizes movement and momentum with restricted motions. Depending upon how joints and shoulders/hips work, it could be potentially devastating for any agile/fast pony to use.

Heck, Tai-Chi is all about maximum force via minimum action/motion. ANY pony could apply those lessons, though I'd imagine the patience would be the killer for practice. No real aggression, so for survival it would seem a poor initial choice. Then again, if they already had some training, the incorporation into a new form would probably give them a decided advantage via long standing discipline. <shrugs> Not my first choice though.

But for simple application, I'm sticking with Judo/Hapkido.

Just my two cents. Love the idea of it though.