My theory on
deer is that if you don't hit them in a way that shocks/breaks their spine, but you still make a very good shot, they will die in 5 seconds. So will a human. The difference is that we flop around on the ground and complain about it (I assume). Their only response is to run like hell no matter what. So in 5 seconds they cover a lot of ground even though they're in the act of dying.
My first deer was a perfect 60
yard shot in the heart. It ran 100+ yards and due to no pumping heart, left no trail (drop every 5 feet on fall leaves). My uncle was a great hunter and tracked it and we found it.
Birds could be a different matter, they do tend to flop for more than 5 seconds even with their head removed.