NO! You do not get near the downed BG. This means that you don't go to kick the gun away, check for pulse, turn him/her over, or anything. You stand close enough to the body (say 7 yards) so that you can watch for further activities by the BG, but not close ehough for the BG to make any moves to shoot, stab, kick, punch or otherwise surprise you. If BG makes any other threatening moves, you shoot him/her again.
Stand nearby and continue to cover with your weapon until LEOs and paramedics arrive, until then, you are still in danger. (There might even be another BG lurking around nearby.)
Unless you are qualified to identify the condition of a person just by looking at him, you are not out of danger from further assault. Leave it up to the police and medicos showing up to render aid.
Many people falsely think that it's over just because someone is shot. But the simple fact is that YOU DO NOT KNOW IF HE/SHE HAS ACTUALLY BEEN SHOT! How big is a bullet hole? Can you really see any bullet holes in dark clothing? Under dark conditions? Sometimes the BG is just unconscious, not hurt bad, or just playing possum. You do not know.
If you have used deadly force to defend yourself, it better be justified, so you do not owe the shot person(s) any obligation at rendering aid until you have proper LEO backup.