I think we may be close to an even split on the bird-stocking aspect here on the forum.
I believe it happens, instinct wise, by a bird like a heron moving live fish.
I don't think the bird egg transport thing is real, but I suppose it has a snowballs chance.
Stocking by human (Ole Mudcat Joe) is the first go-to reason for seeing an odd fish in a body of water, but beyond that, I believe certain birds will move live fish.