For argument sake, think of the running takeoff by certain seabirds as being quite similar to the rolling takeoff of the plane that started all this. The forward propulsion for the bird comes from the action of the wings, not the feet. The feet, despite their flaying of the water surface, do not contribute to the speed of the bird, they just support it until there is sufficient airspeed to take off.
Now, if said bird was attempting to take off on our magically moving lake or a river, and the speed of the water was equal but opposite to that of the bird, the bird would definitely be able to take off. He might get blisters on his feet, but he would get airborne.