I have shot deer everywhere. Brain shots, heart/lung shots, and thankfully only one gutshot. Field dressing a gut shot deer made me swear never to do that again as the insides were just mush and very stinky. I have on more than one occasion (when I was still learning) started tracking the wounded animal as soon as the shot was over, and the deer was out of sight. When a deer is wounded (excluding brain shots which render them unable to run instantly) they WILL sprint off a distance and either bed down or fall over dead. If it falls over dead your lucky.
Depending on the placement of the shot and the vitals injured, the deer may live some time after a questionable shot. Deer will bed down after being wounded as soon as they feel they are far enough away, or too weak to run any longer. If you start the track too early, you run the risk of spooking a dieing deer which may jump up and run another 100yds making the track longer and possibly a longer distance to drag.
What I do when I shoot is try first to focus on the area of impact and the general direction the wounded animal went. Know where it was standing when you shot. Make a mental note. Think pointy rock... strange looking bush.... big oak tree.... right by that group of ferns...etc. Then the deer went over hill over there, by the big rock, or tall tree.. etc... I sit and wait at least 30min, maybe an hour if I have no idea where my shot placement was. (think shotgun slug at 50+ yards) It may be hard to do especially if you can't wait to get your hands on that huge rack!...you must be patient. Give the deer time to die! After at least 30min I then get up. I inspect the point of impact. I look for blood and/or tissue, then start the tracking process. If there is plenty of heavy bleeding, I continue to track. If the blood is spotty with a drip here and there, I may wait another 30min or so. Never immediatly start tracking when the shot is over. (just the way I do it and i consider myself very successful at recovering deer that I shoot.)
If the deer is wounded but still has life left, they will get up and run some more if you happen to run up on him. Remember also: a dead dear will have his eyes open! Deer with eyes closed is not dead yet....