I believe it depends on the scare factor at impact.
If you made a lot of noise, or if you hit a heavy bone, causing more trauma, they run harder to get away from the perceived danger. A deer can run far in just 5 - 10 seconds. IF they are not so scared, they tend to jump away to the first cover, stop and watch, then keel over, or bed and die. All in seconds (assuming a double lung hit).
As above, I have put an arrow thru the chest of several bucks that responded by kicking, looking around, and, in one case, going back to feeding.
Everything eats deer and turkeys. Their normal response to anything out of line is to get out of Dodge.
Normal ? 0 to 100 yards might get most of them.
ChuckC