First off you need to understand what the judges are looking for in those 3 rings and what kind of horse it takes to excel. A hunter classic and a derby are almost the same thing: basically a special hunter class at a show where your first round is a typical hunter round and the second is more handy (will have tight turns, bending lines, hand gallop fences, etc) to test your horse's rideability. All of these, like typical hunter classes are judged on the horse. So to be competitive you need something that moves well, jumps round, has a nice lead change, has a long stride, etc etc. You're being judged on all these things, plus how smooth your round is, how correct your distances, etc.
In the equitation ring the rider is being judged. The course will be more complicated then a typical hunter course but it is still imperative to get to all the correct distances, get your lead changes, be balanced, in control, etc etc. A competitive equitation horse needs to be very balanced and responsive and a lot of people also look for something that has a very smooth trot/canter (makes sitting trot easier!) and flat jump so your position can be correct. It doesn't matter how pretty your horse moves!
A hunter CAN do the jumpers, if it can make the turns and have the speed without knocking down a rail. It's always a plus if the horse can do both rings successfully. It's much harder to find a good jumper that can excel in the hunter ring because often they won't move well or carry themselves like a hunter (long and low). I've shown many a hunter in a jumper class!
If you are wanting to find a horse that will do well in all three rings (hunters, jumpers, and eq) I'd look for a nice hunter that is responsive and balanced and can pick up speed without getting too hot. Like I said, a lot of jumpers won't have the right movement or temperament for the hunters but lots of hunters can make the turns/speed to do well in the jumpers. ANd if they can do jumpers smoothly they can definitely do the eqs too.