I shoot as you described on the vast majority of targets on a sporting course ........ decide where I want to kill it, mount, and back up just a little ways towards the trap. Lift my head ever so slightly from the gun and look back to the place where I can first see the bird after it leaves the trap. As the target gets close to my muzzles I begin a very controlled but short move that keeps my muzzles always ahead of the target. Then when it looks right, kill it. To many, this method looks like you're hardly moving the muzzles at all, but you are. Bill McGuire taught me this method and it has served me very well. I would definitely use it on the presentation you describe. Some of my competitive buddies call it a "controlled ambush". BTW, I never let a fast bird get ahead of my muzzles. If I had to name the style of lead I shoot on most stuff I'd call it pull away, but not started on the bird, rather started from a tiny bit ahead of it. When I kill the first bird in a true pair and make my move to the second one I intercept it's line ahead of it.

The best advice I could give anyone about shooting sporting would be to use the hands less, and the eyes more. Always let the eyes go to the bird before the hands. Where the eyes go the hands will follow. Another Bill McGuire truism.