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We show how, by changing the polarization value of ferroelectric domains, it is possible to tune the magnon conductivity in the ferromagnetic film layer of a multiferroic magnonic system. In particular, we suggest how to switch from a metal behavior (zero frequency gap and linear frequency-wavevector dispersion) to an insulator behavior (around 1 GHz frequency gap and parabolic dispersion). The ferroelectric film is prepared with a sequence of ferroelectric domains with a periodic variation of their polarization direction. Through inverse magnetostriction, they induce in the ferromagnetic layer a periodic magnetic anisotropy and a consequent sinusoidal magnetization. The amplitude of the sinusoidal magnetization can be varied by varying the induced magnetic anisotropy. This allows for a fine and reversible control over the curvature of the dispersion relations at the Brillouin zone boundary, as well as the width of the frequency gap. We suggest the extension of Dirac’s magnon picture to our system, finding interesting implications in terms of magnon mobility. This work expands the possible implementations of the voltage-controlled-bandgap meta-materials, marks the conditions for the occurrence of a magnonic metal behavior in a ferromagnetic film, and outlines how a same unpatterned film can be reversibly turned from a magnonic metal to a magnonic insulator.