IF this is what happened:
Dude assaulted cop and kicked the shit out of him. he was older balding white dude, belly and small arms, didn't look strong. He got face busted up but did guard his weapon. Once brown turned to flee, Cop drew and fired at him, hitting arm, first chance he had.
Brown was pissed, and/or realized at this point, fleeing means you catch 3 more hot ones in the meat of your back, or you double back and take that gun. Either way, he had put himself in that bad spot and ended up with a headshot for it.
Could be what happened. very human and natural reaction to catching an ass whooping - you fire as soon as you can. you don't want to be the cop that got beat down and did nothing, but your face all beat and you can't exactly win if you do catch up with Brown.
*IF* he took a rear bullet. Not sure how else he gets that one. (Unless the hands-up position, I dont even want to go there - hands up changes everything)