  If dark matter self-annihilates into neutrinos or a second component of
("boosted") dark matter that is nucleophilic, the annihilation products may be
detected with high rates via coherent nuclear scattering. A future
multi-ten-tonne liquid xenon detector such as DARWIN, and a multi-hundred-tonne
liquid argon detector, ARGO, would be sensitive to the flux of these particles
in complementary ranges of $10-1000$ MeV dark matter masses. We derive these
sensitivities after accounting for atmospheric and diffuse supernova neutrino
backgrounds, and realistic nuclear recoil acceptances. We find that their
constraints on the dark neutrino flux may surpass neutrino detectors such as
Super-Kamiokande, and that they would extensively probe parametric regions that
explain the missing satellites problem in neutrino portal models. The XENON1T
and Borexino experiments currently restrict the effective baryonic coupling of
thermal boosted dark matter to $\lesssim 10-100 \ \times$ the weak interaction,
but DARWIN and ARGO would probe down to couplings 10 times smaller. Detection
of boosted dark matter with baryonic couplings $\sim 10^{-3}-10^{-2} \ \times$
the weak coupling could indicate that the dark matter density profile in the
centers of galactic halos become cored, rather than cuspy, through
annihilations. This work demonstrates that, alongside liquid xenon, liquid
argon direct detection technology would emerge a major player in dark matter
searches within and beyond the WIMP paradigm.
