Swing through may also be the old 'bum, belly, beak' approach with your mind 'driving' the gun's speed on autopilot, but I'd not suggest it be anything overly conscious when using it or you just missed.
One of the better treatises IMHO on feild technique is Robert Churchill's 'Shooting'. It is a SxS doublegun work and it addresses most field situations, inclusive of taking hare & rabbits from a butt on a driven shoot.
Irrespective of technique, it is visual acquity ... quick target acquisition and intense focus on the target that makes for a good shot. On pheasant and some other game, sex of the bird must be determined before taking the shot. IMO, it is more important to shoot no low birds and know where the other people and dogs are than to become too target intense in the field. On clay targets or the flyers Rocketman alludes to one may be 100% target intense.
I know and have been around some remarkable field shots that don't shoot clay targets at all, never have. In the same breath, I know of no clay target or flyer shooters of any consequence that have not paid their dues with constant practice and working on their games. It's like the old question,"What's the dif between shooting a 98 and shooting a 100 in a tournament?" The correct answer is, "About $20,000 a year."
