The downward force that the aircraft exerts on the air supporting it will increase! Then as time progresses the aircraft will get lighter!
To raise the potential energy of the birds (increased height above the earth) the birds must exert a force on the floor of the aircraft. This requires a thrust downward from the wings of the birds. The birds will increase the altitude of the centre of gravity of the craft. This has the effectiveness of pushing the aircraft downward, the velocity of the downward force will increase its force under the wings of the craft. When the birds reach a constant altitude they will no longer exert this force and the aircraft thrust downward will return to its previous bird seated value,.......
Well not exactly! The birds will have expended heat in the process of raising their potential energy. This heat will have come from chemical energy, so they will be slightly lighter this reducing the weight of the aircraft.
Ever so small changes, but no one said anything about not being precise lol