well at a guess, i reakon the downward force produced by the beating wings would impact on the floor. this force would equal the pigeons weight.
Suppose its similar to riding in a train, when u jump u dont get left behind, u have mass hence inertia
back to the pigeons, once airborne in the cabinu dont think that they will get smacked up against the rear of the aircraft once they are airborne do u?
The mass of the aircraft will remain the same regardless.