This is some great info on here.
I run as fast I can to the shot location. I stare and listen for the slightest sound and then mark the last sound I heard with an arrow or a stick on the ground. If I know I hit em good, or it is getting dark, I will go after them right away. You are sneaking up on an animal that is wheezing and hacking. Even if you get a little closer, you may hear the death tumble or some of that wheezing that you may have missed otherwise. The last 2 elk I got with my bow, I was able to sneak up and find them and wait for awhile for them to expire.
I don't know about deer, but I have seen elk linger around. When you bump elk out in the dark ahead of you, you may have found your animal.
My most important thought:
Don't jump around and high-five your buddies until the job is done. Some A-hole that I hunted with once shot a buck at relatively close range. He turned around and started screaming immediately how "awesome" that shot was. Then we found an arrow. Then we searched for this buck all night and part of the next day. I don't know if his lack of focus made the difference, but it bothers me to think it could have.