  A possible extension of the Standard Model able to explain the recent
measurement of the anomalous magnetic moment of the muon consists in adding a
gauged $U(1)_{L_{\mu}-L_{\tau}}$ symmetry. If the dark matter particle is
charged under this symmetry, the kinetic mixing between the new gauge boson and
the photon induces dark matter-electron interactions. We derive direct
detection constraints on light dark matter charged under a
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry with electron recoil experiments, and
explore prospects with XLZD and OSCURA to close in the parameter space able to
explain simultaneously the recent measurement on the anomalous magnetic moment
of the muon and the observed relic density of dark matter. We further discuss
the spin-dependent scattering contribution arising in this model, which was
ignored previously in the literature.
