Aren't we comparing oranges and oranges?
Tactical shooting includes ALL the factors that Nathan noted. It IS a martial art, though a modern one. Many of the best ways, or at least better ways, to move as human beings are going to be the same or similar.
I had the same experience Nathan did. But I have a friend and veteran of multiple shootings, including one on the move against a laterally moving target, who moves similarly without benefit of martial "art" background. Its still martial movement...
That friend has observed that martial artists seem to pick up moving "properly" with firearms better than others.