Notice that foxhounds are not called "dogs" and their voice or music is not referred to as "barking!" The riders in the hunt follow the fieldmaster (often the Master of Foxhounds himself who is the individual in charge of the entire hunt) in order to try to follow the hounds and observe them if possible without disrupting them. While one hopes to catch a glimpse of the fox the MFH (the Master) is always very careful to try to not impede the animal from its escape nor impede the hounds in their quest. Sometime the hounds move fast and so do the riders! But other times it is about slow methodical work by the hounds, so the horses (and humans) involved must be willing to stand and wait (and enjoy a lovely morning).
There is a lot more to it that the galloping, jumping and "tally ho"-ing one sees in the movies! Horses need to be very fit, surefooted and have very good manners. The ability of both horse and rider to canter or gallop calmly in a group over uneven terrain is something that is very necessary. Given the right introduction many horses can acquire these traits. For this reason LCH and many other hunts offer a second field (that might go slightly slower than first field, with jumps optional) and on some days a third field (which goes slightly slower yet and is a wonderful place to introduce a horse and rider to hunting.)
Traditionally farmers encouraged a hunt to utilize their land and the hunt performed the service of ridding the area of the foxes that killed the farmer's chickens and young lambs. With the change of hunting focus to no longer function as a pest eradication service, but more to preserve traditions and educate people about the ancient sport of hunting with hounds, LCH relies on the cordial relationship with multiple landowners and the Cazenovia Preservation Foundation for the land that it utilizes.