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Do mounted fox hunters in America follow the English riding discipline?
Which horse breed is most common for English-style fox hunting here in america wherever this old sport is still practiced? Can American hunters rent horses (hirelings) for fox hunts?

https://www.youtube.com/watch?v=k2rCB0GpKu4

Do mounted deer hunters in the west follow the western discipline?
Which breeds are most common? American Quarter, Morgan? Grade?

What kind of horseback riding lessons does the novice need to take up if one were to dream of participating in any kind of mounted hunting in the lower 48?

If your interest is fox hunting there are more relevant sites with better answers. A fox hunt can be as demanding as you might want in its style or more moderate. I dont know any western states that allow deer hunting with hounds. There are other locations where deer are sometimes pursued in this manner.
Horsemanship skills and general fitness have to be high to consider these activities. Its far more dangerous to horse and rider in many locales then it is to the actual prey. Probably why its not common. Flat or slight hills with the occasional known obstacles is the only terrain any serious pressure can be safely applied to the prey by horsemen or a reasonable view achieved for the riders.
This is such a varied topic you will only find more accurate details from specific hunts.
My experience is that most horses very naturally respond to the excitement of hound hunts and must be carefully taught whats expected of them . a fox hunt horse will be deadly on a western biggame hunt. And most western horses would require some additional training to foxhunt english style. But i firmly believe its not something many horses would switch between easily either direction. As a rider the experience is also vastly different. If you have a interest in any type of hunting you would be far better served to select one very specific prey style and location for your first experience. And honestly it would take far more then a few months riding lessons to undertake some styles of mounted hunting. Most modern humans dont have the experience or fitness to try pushing any type of boundary with mounted hound hunts. I hunt only on horseback aswell as working cattle in severe terrain on horseback and i am very careful. I also dont assume what skills i might have translate to other styles of mounted hunting equally.