Actually, characters with high speed, endurance, and heavy body weight (body weight helps you carry more, which means minimal equipment weighs you down less) can run down big game like elk, reindeer, and stag. You need to be fully zoomed out, and you need to take walking breaks when you no longer see the animal, but you can run down big game directly. Works best in early spring, when there is some snow on the ground. Enough to leave tracks, but preferably not enough to fatigue you while walking.
High tracking skill helps, as does persistance hunting in pine forests (not spruce forests or spruce mires), and persistance hunting along a river or body of water.
Some animals are generated with better stamina, or speed, but not all of them are. Also, cubs tend to be slower, so if you want to play dirty, injuring the cub and then staying a small distance away, will cause the mother to come back and then run away constantly. She wont (or at least shouldn't) tire herself out this way, but this can be a way to potentially take 2 animals down at once. Cub meat and fur gains have been nerfed in recent versions, but if your character is starving, killing the cub is still a way to deal with your hunger for the next few months, if you preserve it correctly.
Running down animals directly isn't normally how your supposed to hunt though.
Also, I have lost characters while falling out of a tree that I told them to climb. The most gruesome one was a fairly fresh male Kamoulious archer that I rolled up with high stats. I had just finished skinning and butchering a stag and was weighed down to nearly my carry limit for that character. I tried climbing a tree, having forgotten to drop the heavy load, and he fell out of the tree after climbing about like 35, maybe 45 feet up the tree. Landed on his crotch and died instantly.