We describe a general mechanism of controllable energy exchange between waves propagating in a dynamic artificial crystal. We show that if a spatial periodicity is temporarily imposed on the transmission properties of a wave-carrying medium while a wave is inside, this wave is coupled to a secondary counterpropagating wave and energy oscillates between the two. The oscillation frequency is determined by the width of the spectral band gap created by the periodicity and the frequency difference between the coupled waves. The effect is demonstrated with spin waves in a dynamic magnonic crystal.