Can fish be moved by birds - yes. The question is not if but how frequently and how likely that a fish species entered a body of water by this means. I personally saw a GBH, that I surprised, fly off with a catfish, but I cannot tell you if it was in its mouth or speared. Could the bird have dropped it accidentally in another body of water? Sure it could, but how often would that happen. If someone could reliably document a bird purposefully stocking another body of water, I think you could publish your findings in the most premier scientific journals on the planet (Science or Nature). It would seem to me to have very weak evolutionary pressure to evolve such a behavior. I think you could get a Nature or National geogrphic film crew to stake out your location if they believe this behavior is occurring.
