  Coalescing neutron star binary (NSB) systems are primary candidates for
$r$-process enrichment of galaxies. The recent detection of $r$-process
elements in ultra-faint dwarf (UFD) galaxies and the abundances measured in
classical dwarfs challenges the NSB merger scenario both in terms of
coalescence time scales and merger locations. In this paper, we focus on the
dynamics of NSBs in the gravitational potentials of different types of host
galaxies and on its impact on the subsequent galactic enrichment. We find that,
for a $\sim t^{-1}$ delay time distribution, even when receiving a low kick
($\sim 10~{\rm km~s^{-1}}$) from the second supernova explosion, in shallow
dwarf galaxy potentials NSBs tend to merge with a large off-set from the host
galaxy. This results in a significant geometrical dilution of the amount of
produced $r-$process elements that fall back and pollute the host galaxy gas
reservoir. The combination of dilution and small number statistics produces a
large scatter in the expected $r$-process enrichment within a single UFD or
classical dwarf galaxy. Comparison between our results and observed europium
abundances reveals a tension that even a systematic choice of optimistic
parameters in our models cannot release. Such a discrepancy could point to the
need of additional $r$-process production sites that suffer less severe
dilution or to a population of extremely fast merging binaries.
