  The measured branching fractions of $B$-mesons into leptonic final states
derived by the LHCb collaboration hint towards the breakdown of lepton flavour
universality. In this work we take at face value the so-called $R_{D^{(*)}}$
observables that are defined as the ratios of neutral $B$-meson charged-current
decays into a charged $D$-meson, a charged lepton and a neutrino final state in
the tau and muon channels. A well-studied and simple solution to this charged
current anomaly is to introduce a scalar leptoquark $S_1$ that couples to the
second and third generation of fermions. We investigate how $S_1$ can also
serve as a mediator between the Standard Model and a dark sector. We study this
scenario in detail and estimate the constraints arising from collider searches
for leptoquarks, collider searches for missing energy signals, direct detection
experiments and the dark matter relic abundance. We stress that the production
of a pair of leptoquarks that decays into different final states (i.e. the
commonly called "mixed" channels) provides critical information for identifying
the underlying dynamics, and we exemplify this by studying the $t \tau b \nu$
and the resonant $S_1$ plus missing energy channels. We find that direct
detection data provides non-negligible constraints on the leptoquark coupling
to the dark sector, which in turn affects the relic abundance. We also show
that the correct relic abundance can not only arise via standard freeze-out,
but also through conversion-driven freeze-out. We illustrate the rich
phenomenology of the model with a few selected benchmark points, providing a
broad stroke of the interesting connection between lepton flavour violation and
dark matter.
