The Milky Way's stellar disc can tilt in response to torques that result from infalling satellite galaxies and their associated tidal debris. In this work, we explore the dynamics of disc tilting by running N-body simulations of mergers in an isolated, isotropic Milky Way-like host galaxy, varying over satellite virial mass, initial position, and orbit. We develop and validate a first-principles understanding of the dynamics that govern how the host galaxy's stellar disc responds to the satellite's dark matter (DM) debris. We find that the degree of disc tilting can be large for cosmologically motivated merger histories. In particular, our results suggest that the Galactic disc may still be tilting in response to Gaia-Sausage-Enceladus, one of the most significant recent mergers in the Milky Way's history. These findings have implications for terrestrial direct detection experiments as disc tilting changes the relative location of the Sun with respect to DM substructure left behind by a merging galaxy.