Another thing to consider is, the longer you wait, the more likely the wound will plug up and there will be no blood trail to follow.
Back east where this is popular, you push an animal and it goes onto some other property you don't have permission to hunt. So they hope by not pushing it, it will die on the property they can hunt.
Give it a few minutes to settle down, cuz it doesn't even know what happened. Then follow the blood trail slowly just like you're still hunting and if need be, get another shot into it.
If it's dark, this is not only difficult but also illegal.
Edit: Shooting it again after dark would be illegal, not following it.
Also, by your own logic of the bleeding slowing and/or stopping over time, doesn't this make it all the more important you don't bump it out of it's bed? I've learned this the hard way.