  We consider the generic possibility that the Universe's energy budget
includes some form of relativistic or semi-relativistic dark radiation (DR)
with non-gravitational interactions with Standard Model (SM) particles. Such
dark radiation may consist of SM singlets or a non-thermal, energetic component
of neutrinos. If such DR is created at a relatively recent epoch, it can carry
sufficient energy to leave a detectable imprint in experiments designed to
search for very weakly interacting particles: dark matter and underground
neutrino experiments. We analyze this possibility in some generality, assuming
that the interactive dark radiation is sourced by late decays of an unstable
particle, potentially a component of dark matter, and considering a variety of
possible interactions between the dark radiation and SM particles.
Concentrating on the sub-GeV energy region, we derive constraints on different
forms of DR using the results of the most sensitive neutrino and dark matter
direct detection experiments. In particular, for interacting dark radiation
carrying a typical momentum of $\sim30$~MeV$/c$, both types of experiments
provide competitive constraints. This study also demonstrates that non-standard
sources of neutrino emission (e.g. via dark matter decay) are capable of
creating a "neutrino floor" for dark matter direct detection that is closer to
current bounds than is expected from standard neutrino sources.
