  Thermal dark matter at the MeV mass-scale has its abundance set during the
highly non-trivial epochs of neutrino decoupling and electron annihilation. The
technical obstacles attached to solving Boltzmann equations of multiple
interacting sectors being both relativistic and non-relativistic have to-date
prevented the full treatment of this problem. Here, for the first time, we
calculate the freeze-out of light dark matter, taking into account the energy
transfer between the dark sector, neutrinos, and the electromagnetically
interacting plasma from annihilation and elastic scattering processes alike. We
develop a numerically feasible treatment that allows to track photon and
neutrino temperatures across freeze-out and to arrive at a precision prediction
of $N_{\rm eff}$ for arbitrary branching ratios of the dark matter annihilation
channels. In addition, our treatment resolves for the first time the dark
matter temperature evolution across freeze-out involving three sectors. It
enters in the efficiency of velocity-dependent annihilation channels and for a
flavor-blind $p$-wave annihilation into electron- and neutrino-pairs of all
generations, we find the present Planck data excludes a complex scalar dark
matter particle of mass of $m_\phi \leq 7$ MeV.
