Dense nuclear matter in the core of neutron stars consists, in its simplest manifestation, of a plasma of degenerate neutrons, protons, electrons, and muons in $\beta$ equilibrium. I present a calculation of the scattering rates of electrons and muons, taking into account correlations of strong and electromagnetic interactions. These correlations allow for medium-induced lepton-neutron scattering, which in turn leads to a sizeable modification of the total scattering rate. The impact of induced interactions is particularly pronounced close to the crust-core boundary of neutron stars, which makes the results potentially relevant for the damping of hydrodynamic modes and r-modes.