so today i caught one of the contestants on my patio with a mouse. a live mouse, trying to run away. so i said "all right. that's enough. this is MY patio and that is MY mouse."
and by this time the mouse had been chased under the fence and was lying exhausted in the grass on the other side. so i turned my best prey stalking manner on the cat. after a few rounds of "i'm standing crouched and motionless while you're looking at me and when you look away from me i move closer" he turned and ran.
then i turned back to the poor mouse, still lying exhausted on the grass. i wriggled under the fence on my belly to get to her. she looked ok, but she'd peed herself. i picked her up gingerly, and she didn't have the strength to bite.
out in the open is a bad place for an exhausted mouse, so i carried her back and put her in a hole in my woodpile. mice nest in there sometimes. i don't mind. mice are not allowed to nest in the house, but the woodpile is ok.
she rested there for a while and in an hour or so she was gone.
mice, left to rest, recover pretty fast. voice of experience.