If all the birds were in synch then the weight momentarily varies up and down, but the average weight over time stays the same.
However the interesting question is whether the floor is not solid but say, cage-like. When landed the birds weight transfers in total to the vehicles frame, but in the air much of the downforce goes through to press on the road below. So what about the total weight then ?
Unsure about that one vascop. Planes stay aloft due to the speed of airflow over and under the wing, rather than flapping. My suspicion is that the weight doesn't change for either vehicle.