There are a couple of problems with trying to generalize about the effect of bore diameter on patterns. Roundball has note one of them. He is correct. Its easy to mistake a " good pattern " when shooting at a stationary pattern board target from a standing or bench rest. Those patterns are fine for shooting STATIONARY targets.
The problem that affects flying targets is the fact that the smaller bore ( 20 ga.) is going to Require the same amount of shot to occupy a longer area of the barrel, which means that more pellets in the load of shot are rubbing against the bore of the shotgun barrel. When the gun is fired, those pellets all rub against the barrel going out the barrel, leaving flat spots on the pellets, and lead deposits in the barrel.
The flat spots on the pellets cause these pellets to slow down quickly, and fall out of the pattern, usually within the first 20 yards. The smaller the gauge, or bore diameter, the more shot is lost from the measured shot load, in that first 20 yards.
The lead deposits also adversely affect pattern performance, because they act like speed bumps on a roadway to the next shot load fired out the barrel. Worse, they cause even more lead to be deposited in the barrel making more and higher speed bumps. This makes for more and more ragged shaped patterns, and can even change the POI for the entire pattern when enough lead deposits build up in the barrel.
In general, then, you would be better off using a 20 gauge load in a large bore shotgun, say a 12 gauge, and achieve a better pattern for flying targets, than shooting that same load of shot in your 20 gauge.
For stationary targets, you are interested in the number of pellets you can put in the core area of the pattern. You don't care about the fringe, or edges of the patterns. On Moving targets, you want an even, round pattern , to maximize your chance to hit a bird flying through the air.
No bird will teach you faster this important consideration than the morning dove, which literally can turn on a dime, and give you back 9 cents in change! And doves have an uncanny ability of changing direction just as you are slapping that trigger! That is why we use open choked guns( cylinder bore Muzzleloaders are great) at relatively short ranges,with shot charges consisting of very small shot ( #8 or even #9) so that we can fill the air with lots of pellets to have a better chance to hit the bird before he dives under our shot.
So, if you are using a shotgun to hunt deer, or turkeys, only, then use that 20 gauge gun with the longer shot column to concentrate more shot in the core area of the pattern. If, on the other hand, you are looking for a smoothbore to shoot winged game, choose the larger gauges, but shoot lighter charges of shot when hunting smaller birds. A 12 gauge, shooting a 28 gauge load of shot( 5/8 oz.)for instance, will actually throw a better SHAPED pattern, with more pellets in the pattern, than if you shot that same load from a 28 gauge smoothbore.
Can you improve the pellet count in your 20 gauge patterns? Yes-- By using shot cups-- paper, or plastic-- and by lubing the bore of the shotgun in front of the load, so that the lead pellets slide over the bore rather than rub against bare steel.
Will those changes also improve the shape of my patterns using that 20 gauge? Yes, but the same load in a larger bore shotgun will usually still do better.
