I will present and compare several cosmological constraints on the cross section for scattering between dark matter (DM) and baryons, for cross sections with non-trivial dependence on the redshift or relative velocity. I will demonstrate a general framework for estimating the cosmic-microwave-background constraints on general scattering histories, and discuss the implications of these constraints for observations of the redshifted 21cm line from the cosmic dark ages. I will also present some new results on the potential for early-universe observations, and indirect searches more generally, to observe the effects of 3-body annihilation processes, particularly in the presence of a potentially large low-velocity enhancement to the DM annihilation rate.