  Dark Matter (DM) models providing possible alternative solutions to the
small- scale crisis of standard cosmology are nowadays of growing interest. We
consider DM interacting with light hidden fermions via well motivated
fundamental operators showing the resultant matter power spectrum is suppressed
on subgalactic scales within a plausible parameter region. Our basic
description of the evolution of cosmological perturbations relies on a fully
consistent first principles derivation of a perturbed Fokker-Planck type
equation, generalizing existing literature. The cosmological perturbation of
the Fokker-Planck equation is presented for the first time in two different
gauges, where the results transform into each other according to the rules of
gauge transformation. Furthermore, our focus lies on a derivation of a broadly
applicable and easily computable collision term showing important
phenomenological differences to other existing approximations. As one of the
main results and concerning the small-scale crisis, we show the equal
importance of vector and scalar boson mediated interactions between DM and
light fermions.
