I don't have optics on my shotguns so I'm happy to be corrected but, I believe they'd work just fine for static targets, but on dynamic targets I just don't think they'd work.
Birds on the wing or clays, I use my front (pump hand) thumb to lead the bird. From my eyes perspective the thumb is in between the front bead and the bird (or clay). It's how I was taught to lead and it works every time if I do my part. With a red dot, I'm happy to understand the dot replaces where the bead would be, but how do i see my thumb? Won't the red dot housing get in the way of that sight picture?
Said another way, I don't believe red dots can be used to lead a target that's moving, and that's a big problem for a fowler.