"Situation dependent" is what I agree with. For some mil/Le situations, you might need to keep moving while shooting to let the rest of your team enter and engage. If a person is engaged by multiple attackers at close range, being a moving AND fighting target may make the difference.
In a 1 v 1 encounter at short to mid range, I think most people would be better served with a fast and accurate (if stationary) response. I know that, standing, I can deliver a pretty good volley of accurate fire in a short time. Moving either cuts the accuracy or speed of the shots. In stopping any particular attacker, delivering multiple torso/head shots in a short time seems to be the winning combination more often than not.