Laws and locations vary, but ...
Reduction to possession
Edited to add:
Which is to say, know your laws, because if you come upon “your” game animal, and someone else has reduced it to possession already, there is a better than average chance that the other party has LEGAL rights to that downed animal ... hunter’s ethics aside.
I can’t speak for other states, but in Montana, the killing shot is the one who is supposed to tag the animal…