You can do just fine with a 20ga. My daughter has a Mossberg 500 youth model that shoots the Hevi-13's out of a Jellyhead choke tube about as well as you could hope. My brother took a couple birds a few years ago with a Winchester SX3 in 20 ga. I think he tried to stretch the range a bit cause he missed one bird and had to shoot another one twice.

Just get 'em in close (30 yards or less) and you will have no problems.