Abstract Quasar lines of sight intersect intervening galaxy discs or circumgalactic environments at random impact parameters and potential well depths. Absorption line velocity widths (Δv90) are known to scale with host galaxy stellar masses, and inversely with the projected separation from the quasar line of sight. Its dependence on stellar mass can be eliminated by normalizing with the emission-line widths of the host galaxies, σem, so that absorbers with a range of Δv90 values can be compared directly. Using a sample of DLA systems at 0.2 <z < 3.2 with spectroscopically confirmed host galaxies, we find that the velocity ratio Δv90/σem decreases with projected distances from the hosts. We compare the data with expectations of line-of-sight velocity dispersions derived for different dark matter halo mass distributions, and find that models with steeper radial dark matter profiles provide a better fit to the observations, although the scatter remains large. Gas outflows from the galaxies may cause an increased scatter, or scale radii of dark matter halo models may not be representative for the galaxies. We demonstrate by computing virial velocities, that metal-rich DLAs that belong to massive galaxy haloes (Mhalo ≈ 1012 M⊙) mostly remain gravitationally bound to the haloes.