The diffuse supernova neutrino background (DSNB) offers a unique way to probe MeV-scale dark matter (DM) by boosting DM particles in the Milky Way to detectable energies. Considering minimal models of DM-electron and DM-neutrino interactions, we study energy-dependent scattering cross sections and analyse DM attenuation due to electron scattering in the Earth. New bounds on these interactions are set using electron recoil data from XENONnT. We show that energy-dependent cross sections can significantly alter previous constraints based on energy independent cross sections, highlighting the need for studies on a model-by-model basis.