The best advice is like Westtex said, drop the deer in it's tracks. BUT that doesn't always happen.
My grandpa shot a doe in the rain last season, caught it a little far back and we knew it was a bad hit. Being that he's in his late 70's it was my job to track her. It was raining hard, and I knew any blood trail wouldn't last long. So, I decided to get on it right away in hopes of "catching up" with the deer. I was on the blood trail about a minute after he shot.
He went back to the truck, and drove to the other side of the property to watch in case the deer came out that way. I followed the blood and the turned up leaves slowly for about a hundred and twenty yards. I was scanning out in front hoping to see the deer first. FInally I saw her bedded down laying under a fallen tree. She had her head up looking around. One shot in the neck and it was all done. The blood was quickly washing away and had we waited for it to just lay down and die we probably never would have seen it.
So, the lesson I learned that time was that if the blood trail is gonna wash away, you are gaining nothing by waiting long enough for the deer to lay down. BEst advice is to get on the trail right away and hope to catch the deer bedded down.
-jake