I've found that you should run as little as possible. WALK when you follow your prey. When they start to get slightly out of breath I have my character run when I see it until the animal starts running, and then return to walking, with the aim to keep the animal run as much as possible, ending up in them totally exhausted, with the character recovering between short sprints while walking.
IF you manage to hit the target with a ranged attack that will slow it down, or at least get it to tire faster (in particular if it becomes lame), but hits are far from guaranteed.