In shotguns...where we are shooting at flying targets....we need to pull the trigger quickly / so a lot of us refer to it as "slapping" the trigger....but we don't mean "jerk" or "yank" either....its a controlled "slap"...but the other issue on a shotgun is the shot is only starting as you pull the trigger - and we have to follow thru by continuing to move the gun as the target breaks ...but we move it smoothly .../ so in my mind, its a very different technique than pulling a handgun trigger. We shouldn't change the direction of the shotgun ...when we pull the trigger....if you do, then you need more control of trigger pull - and maybe a better mount. In shotgun shooting ...the gun is anchored in our shoulder ...and in both hands / in a good stance...the lead hand is only stabilizing the gun - not moving or steering the gun...but with an "anchored" shotgun, executing a trigger pull should not move the muzzle of the shotgun.
In handguns - especially in bulls eye practice....we squeeze the trigger ( moving only the trigger finger ) we don't want to "milk" the grip on the gun, we only want to put pressure on the trigger, and increase it steadily and slowly - but emphasis on steadily -- with the gun on target, and the gun should fire as a surprise.
That technique will not work on shotguns.../ on shotguns we need a "controlled slap" of the trigger...think "smooth".....but focus on just firing the shotgun with your trigger finger...don't milk the grip of the shotgun either....
at least that's my perspective....I'm not an instructor / just a shooter ...and over time, I've developed techniques that work for me.
Flinches.....in my opinion....are vision related / where a shotgun shooter looks at the bird ---then the barrel -- then the bird ( and he flinches )...the trick is, never look at the barrel....the only thing you look at is the leading edge of the flying target ( not the butt / not the "whole target" ..just the leading edge ....and you don't "measure" the lead....you "feel" the lead ....pull the trigger and keep the shotgun moving and follow thru....
If you're having trouble on follow thru ....I suggest shooting more Skeet...( especially pairs )....because it forces you to follow thru, then shift your eyes, find the 2nd target...then move the gun ..and execute a shot on the 2nd bird in the air .....and then take your successes to the Trap field.