You see, any herbivore has to spend a large amount of time simply eating, and grass, leaves, etc. don't provide a lot of nutrition. So it takes a lot of food, and the deer or horse has to spend a lot of time storing up the energy it takes to run away from the hunter.
All a person has to do is stay close enough to the animal to prevent it from having the time to eat as much as it needs, or peace enough to rest and digest. They only need to be disturbed once or twice a day, at first, while they are still strong.
Of course, the process might take a week!
For that matter, a person can outrun (stay ahead of) a horse with a rider, given really good conditioning and an 8 to 10 hour head start. A rider simply adds 200 pounds or more to the food and rest problems noted above.