What makes a flying object keep moving with the Earth's rotation? When a bird is flying or when a copter is hovering or a plane is flying what force is making them rotate with the earth beneath them?
They aren't.  To first order, airborne objects are moving with the Earth because they were moving when on the ground.  However, what they retain is their angular speed, and since the circumference increases with altitude, an object that doesn't accelerate appropriately will appear from the ground to fall behind to the West (the Earth is rotating to the East).  This is a well-known phenomenon whose magnitude is latitude-dependent, falling to zero at the poles.