Here is one test done by others
Pattern results that compare two similar 1-ounce #7 ½ lead factory loads shot through 20-gauge and 12-gauge guns. They aren't exactly the same load (not sure how you could do that anyway), but close enough to show the patterning differences of the two gauges (these two guns anyway). The patterns are the average of five shots, 30" post-shot scribed circle, yardage taped muzzle to target, and the in-shell pellet count is the average of five shells.
20-gauge Browning Invector-plus w/ 28" barrel and Briley flush chokes
REM SURESHOT 1 oz #7 ½ lead (349 pellets) @ 1,165 fps
30 YARDS
CYL / 179 (51%)
SK / 221 (63%)
IC / 249 (71%)
12-gauge Browning Invector-plus w/ 28" barrel and Briley flush chokes
WIN AA XTR-LITE 1 oz #7 ½ lead (344 pellets) @ 1,180 fps
30 YARDS
CYL / 194 (56%)
SK / 248 (72%)
IC / 288 (84%)
In this pattern test, the larger 12 bore showed it was more “efficient” (about 5-10%) than the 20 bore. As you can see from the pattern numbers, it took an IC choke in the 20-gauge to match the performance level of the SK choke in the 12-gauge.
Many factors influence patterning, but I believe this patterning shows the general principle of larger bores being more “efficient” than smaller bores. However, the pellets may or may not be of the same quality (antimony content) which could affect patterning and allow the 12-gauge to shoot this particular load better than the 20-gauge.
In practical terms, the effectiveness exhibited with the same payload and shot size between 20- and 12-gauge bores would be insignificant as long as they were both choked properly to attain similar pattern numbers at the distance they were used. That said, it looks as though the larger bore can usually do that with less choke constriction.