@ Uppercut, you've whittled down my point into an absurdity. Obviously he wasn't standing still but the kind of movement I am talking about in those headers in question weren't of a nature where I'd be saying "well, Carroll would've never made that run" for my point that Carroll would have finished those chances not to matter. I am considering his movement but it wasn't the kind that Carroll - or any forward at that level worth his salt - wouldn't do.
Take Rooney's miss-header from Young's cross, it is a nice, slightly late, run in the box but it's a stretch to call it some run. Carroll would've done the same and probably would've buried the chance. Or the Gerrard free kick which he half-assedly connected with. It's a two-metre straight run in the middle of the box. Although, Carroll was there on that play - far post, I think. On both he could've done far better and to excuse it as if somehow he made the chances with his intelligent running is just an underserving defense of his ineptitude.
It's speculation, subjective and arbitrary but to reiterate a bit more clearer: given the starting and finishing places of Rooney in those instances, I'd expect Carroll to not only make those runs but to head them far better. Also, Carroll doesn't stay up all the time. As a Liverpool fan, sometimes I wish he would. He's often come too deep and not made the long run in the box for there to be someone to aim to.
A correction: the Johnson cross he actually had very good movement. I must have been thinking of somebody else.