Despite groundbreaking observations of supersolidity in spin-orbit-coupled Bose-Einstein condensates, until now the dynamics of the emerging spatially-periodic density modulations has been vastly unexplored. Here, we study the dynamic behavior of density stripes in such a supersolid condensate subject to spin perturbations. We show both analytically in infinite systems and numerically in presence of a harmonic trap how spin waves affect the supersolid's density profile in the form of crystal waves, inducing oscillations of the periodicity as well as the orientation of the fringes. Both these features are well within reach of present-day experiments. Our results show that this system is a paradigmatic supersolid, featuring superfluidity in conjunction with a fully dynamic crystalline structure.