The one thing I have heard and seen most often in serious wounds is the grace period the body offers for the "fight or flight" instinct to kick in, being a sudden release of endorphins that blunt the pain and give a boost in strength. Wounded people are capable of some pretty impressive actions for at least a moment or two. I've covered cases where people with clearly fatal wounds that resulted in the kind of massive blood loss that would result in death in just moments managed to run or drive a block or two, run into a house, etc.
The only sure put-him-down-now wound is to the central nervous system which destroys a great deal of the gray matter and results in essentially instant unconsciousness. Yes some folks with lesser wounds will curl up and whimper, but overall most will at least try to escape to some real or imagined place of shelter or safety.
I've also seen people with what looked like minimal wounds be talking normally one moment and collapse and die the next, often from an internal bleeder. A puncture of the femoral artery will often do this as blood pressure suddenly drops and down they go. I remember one guy who called the police to complain about the guy who had threatened him with a pistol and actually fired a shot into the ground in front of him. He did not even know he had been hit by the ricocheted round in the thigh until he turned pale and dropped and was DOA at the ER.