Yeah, but what if the aircraft is C-5 or similar? Takes off with the pigeons all strapped in, and once airborne, the birds all take off. Previous answers would say the weight doesn't change.
Then they open the front cargo door (ignoring the real facts like "impossible" and "pilot can't see") and the rear door too. The birds are flying, with air passing through the cabin. If one flies out the back door, has the weight changed?
If a million pigeons in a tight flock now flies through the cabin from one door to the other, does the plane suddenly drop from the sky due to the extra weight?? And then zoom back up when they flap out the back door?
How did Superman fly without flapping his arms or otherwise displacing air downwards?
