This depends on the type of horse. Some horses will be very forward and responsive but other horses will be slow and will plod along. They will mainly walk or trot in the field but will trot/canter if spooked or exited
cross country, show jumpig, point to point, racing . . . that's all i can think of ???
I have never heard of horse events requiring vaccines to be given to horses entered in them. The only related item that most competitions require is a current (within the last year) negative coggins test. A coggins test checks for the presence of Equine Infectious Anemia which is a highly contagious disease in horses. A coggins test is not a vaccine, only a test for the disease. There is no cure for the disease and horses that test positive for EIA must be euthanized or kept quarantined from other horses for the rest of their lives.