We study galactic enrichment with rapid neutron capture (r-process) elements in cosmological, magnetohydrodynamical simulations of a Milky Way-mass galaxy. We include a variety of enrichment models, based on either neutron star mergers or a rare class of core-collapse supernova as sole r-process sources. For the first time in cosmological simulations, we implement neutron star natal kicks on-the-fly to study their impact. With kicks, neutron star mergers are more likely to occur outside the galaxy disc, but how far the binaries travel before merging also depends on the kick velocity distribution and shape of the delay time distribution for neutron star mergers. In our fiducial model, the median r-process abundance ratio is somewhat lower and the trend with metallicity is slightly steeper when kicks are included. In a model 'optimized' to better match observations, with a higher rate of early neutron star mergers, the median r-process abundances are fairly unaffected by kicks. In both models, the scatter in r-process abundances is much larger with natal kicks, especially at low metallicity, giving rise to more r-process enhanced stars. We experimented with a range of kick velocities and find that with lower velocities, the scatter is reduced, but is still larger than without natal kicks. We discuss the possibility that the observed scatter in r-process abundances is predominantly caused by natal kicks removing the r-process sources far from their birth sites, making enrichment more inhomogeneous, rather than the usual interpretation that the scatter is set by the rarity of its production source.