I did a highly non scientific but interesting test a few years back shooting Sporting Clays with a .410.
I'd shoot the birds on a station with a 2-½" shell, then a 3". For the next station I'd reverse it and so on, all the way to Station 12.
I did it for a couple of months, once a week. The object was to see if the 3" gave me better breaks or more hits.
I found no visible difference in hits or breaks with the 2 shells and have shot nothing but 2-½" shells at Sporting Clays ever since.
Since I reload .410, I got a glimpse of what I think is the reason for there being little to no difference. If you reload a 2-½" shell, after you put the wad in and drop the shot you see that the shot is very close to the top of the wad. If you reload 3" shells, you use the same wad, but 1/4 oz. more shot (or 3/16). What happens is the additional shot is above the wad and has no protection from friction from the barrel. I believe that a lot of the shot above the wad gets honed down to where it's about like a frisbee and kind of peels off into never never land when it leaves the barrel. Obviously not all of the shot does as some is still protected but a lot has this happen.