I think for the short distances Cecil speculates eggs being transported successfully, live fish transport is just as possible if not more likely.
1. I have seen a GBH catch a one poundish LMB, then fly to the other side of the pond. I enticed the GBH to drop the bass (unfortunately on dry land) and fly away. The fish flopped around for much of the time it took me (from a couple hundred yards away)it took me to make my way to him, but was beyond reviving when I got there. There were no marks on the bass.
2. We have had a couple of small (3" - 4") BG show up in our horse barn, 400 yards from our pond. I don't know if birds brought them; they didn't have any marks and no one fessed up to leaving them there. I really, really doubt they were brought by a human.
3. I have seen film of crows passing a small fish back & forth in flight, apparently for fun.
"Live like you'll die tomorrow, but manage your grass like you'll live forever." -S. M. Stirling