Magnons, the quanta of the oscillations of localized electron spins, are a powerful tool for information transport and processing of microwave signals. Owing to the challenge of energy efficient spin-wave control on small time-and space scales, dynamic magnonic crystals have been proposed. Their distinct feature is the possibility to toggle on and off the spatial periodicity of the magnetic waveguide that allows one to realize the unusual signal processing functions. The miniaturization of magnonic circuits, reduction in energy consumption, and fast operation are important possibilities of these artificial crystals. These can be achieved in ferrite-ferroelectric (multiferroic) heterostructures, where strong coupling of magnons and microwave photons constitutes quasiparticles called electromagnons. Using both a theoretical approach and microwave measurements, we report on successful dynamic control of electromagnonic band structures in artificial thin film crystals via application of a voltage to the grid electrode located on a ferroelectric film.A promising functionality of the proposed waveguiding structures arises from two major factors: (i) low energy consumptions due to the thin ferroelectric layer, and (ii) pronounced rejection bands caused by a gradual change of the dielectric permittivity.