The magnetic properties of a magnetic material can be modified by elastic deformation—termed the magnetoelastic effect. This effect is considered an alternative approach to magnetic fields for the low-power control of magnetization states of nanostructures since it avoids charge currents that create heat dissipation. This article describes the effects of dynamic strain accompanying a surface acoustic wave on magnetic nano-elements. We use a technique based on stroboscopic x-ray microscopy to simultaneously image the evolution of both strain and magnetization at the nanometer length and picosecond time scales. The study shows that there is a delayed response of the magnetization to dynamic strain, adjustable by the magnetic properties of the material. The presented analysis provides insights into dynamic magnetoelastic coupling in nanostructures with implications for the design of strain-controlled nanodevices.