Diffuse neutrinos from past supernovae in the Universe present us with a unique opportunity to test dark matter (DM) interactions. These neutrinos can scatter and boost the DM particles in the Milky Way halo to relativistic energies allowing us to detect them in terrestrial laboratories. Focusing on generic models of DM-neutrino and electron interactions, mediated by a vector or a scalar boson, we implement energy-dependent scattering cross-sections and perform detailed numerical analysis of DM attenuation due to electron scattering in-medium while propagating towards terrestrial experiments. We set new limits on DM-neutrino and electron interactions for DM with masses in the range MeV, using recent data from XENONnT, LUX-ZEPLIN, and PandaX-4T direct detection experiments. We demonstrate that consideration of energy-dependent cross-sections for DM interactions can significantly affect constraints previously derived under the assumption of constant cross-sections, modifying them by multiple orders of magnitude.