I think that's a matter of whether you're doing straight boxing types of movement or whether you've got the ability to adapt your tools to the game you're playing. Command of range plays a big role, as does one's ability to use the movement of the upper body (a much more accurate way to describe "head movement") for better applying offensive tools. To be sure, just moving around and trying to avoid blows is silly. But a good fighter uses his upper body movement to set up strikes and hide angles. When that's combined with good offense and command of the ring/range, it can be really effective.
I'll look for some coaches who teach it well, but I have to be honest, I don't know of anyone off the top of my head. I just don't follow it like I used to.
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