Well, all 3 disciplines (hunter, jumper, and foxhunter) are English riding disciplines involving jumping, so there's no reason a horse could not participate in all 3. Any horse that is safe and honest over fences could participate in any of these disciplines, but it would be very good to find a horse that is GOOD at all three.
The foxhunter needs to be brave and tough. He needs to follow the hounds wherever they lead: over hill and dale, through streams, over any fences that get in the way, across roads, past scary things, basically willing to go anywhere to follow those hounds. In my experience foxhunting, you do a lot of galloping like an idiot over some really bad footing, then spend long periods just standing around waiting for the hounds to find the scent again, they take off galloping blindly again. The horse needs to be brave, trustworthy, clever, and very sound. Foxhunters have to work on whatever footing they travel over, be in mud, concrete, rocks, etc. Foxhunting horses do not need to be pretty. They can be poor movers, lack style, display poor form, or just be plain ugly. They just need to do their job, not be pretty. The fox hunter must be safe because the jumps are solid and will not fall down if the horse hits them, unlike the jumps you find in an arena.
The jumper also needs to be brave and fast. They are only judged on whether they go clean, so it does not matter if they are not pretty. Like the foxhunter, they can be ugly or have poor form. It does not mater if their knees are even, they can even hang a leg over fences as long as they don't knock anything down. Fences in the jumper ring can be tricky, so the horse needs to be clever and careful as well as fast.
While jumpers and foxhunters only need to be able to get over the jumps, the hunter has to do it with STYLE! The hunter is not judged simply on getting over the jumps; the judge considers his way of going. His canter should be rhythmic and he should be a good mover. He should always find the perfect "spot" to leave the ground before every fence, never jumping too long or coming in too close to the base of the jump. He should display a perfect bascule over the fence and knees should be tight and even. The perfect hunter should appear pleasant and pleasurable to ride. While you see many jumper horses fighting their riders trying to go faster to the jump, the hunter should hold a consistent pace and always look obedient to the rider. While the hunter must be showy, the jumps themselves are a bit more straight-forward than those the jumper must face. They do not get as high, and the courses are less complicated. Unlike the foxhunter, the show hunter gets groomed arena footing to work on.
Theoretically, a horse that is pleasant and obedient to ride, moves well, and and jumps with perfect form should be able to perform any sort of jumping discipline. But where the hunter is smooth and consistent in his pace, he may not be able to handle the pressure of faster speeds and more difficult courses that the jumper faces. And he may not be tough and brave enough to foxhunt. Plus, a good show hunter is worth enough money that you don't want to risk injuring them with foxhunting on difficult terrain.
Hope this helps.
Answered By: Ang - 7/16/2009