If you shoot well forward, none of this happens. You may not have so much game down, but each one of them drops stone dead without a sound. There is no calling out, “Bring a dog, I have a ‘runner.’”
I think it would be as well, before trying moving shots with a pistol, to do a little shotgun shooting at clay pigeons, so as to get into the idea of swing and timing, if you are not a shotgun shot already.
When you can swing your gun to an imaginary spot, in front of a moving object and press the trigger at the moment the sights are aligned, without stopping your swing, you can shoot the pistol with success at moving objects, provided you treat it exactly as if you were using a shotgun.
Have a moderately large object which the bullet will either break or leave a visible hole through, arranged to pass you at a slow speed.
It can either be dragged by a long string, run on a trolley (the trolley shielded behind a bank so that a bullet could not strike it) or some other slowly moving target.
A swinging object is of no use. It makes a difficult curve to follow, for the beginner, and its passage lasts too short a time.
A swinging object also makes the shooter try the objectionable method of waiting and aiming at the spot the object swings to, which I want to avoid.
If your target travels slowly enough, and is large enough, and at only some twelve yards’ distance, there will be no necessity to aim in front of it. Its forward edge is far enough.
Fix your eyes on the front part of the target. As it traverses bring your pistol up without looking at the pistol, as it comes level with your eye and the sights get aligned. Keep on swinging your body and pistol and press the trigger, while still swinging.