  We analyze the prospects for detection of light sub-GeV dark matter produced
in experiments designed to study the properties of neutrinos, such as
MiniBooNE, T2K, SHiP, DUNE etc. We present an improved production model, when
dark matter couples to hadronic states via a dark photon or baryonic vector
mediator, incorporating bremsstrahlung of the dark vector. In addition to
elastic scattering, we also study signatures of light dark matter undergoing
deep inelastic or quasi-elastic NC$\pi^0$-like scattering in the detector
producing neutral pions, which for certain experiments may provide the best
sensitivity.
  An extensive appendix provides documentation for a publicly available
simulation tool {\tt BdNMC} that can be applied to determine the hidden sector
dark matter production and scattering rate at a range of proton fixed target
experiments.
