  We perform an improved cosmic microwave background (CMB) analysis to search
for dark matter-proton scattering with a momentum-transfer cross section of the
form $\sigma_0 v^n$ for $n=-2$ and $n=-4$. In particular, we present a new and
robust prescription for incorporating the relative bulk velocity between the
dark matter and baryon fluids into the standard linear Boltzmann calculation.
Using an iterative procedure, we self-consistently include the effects of the
bulk velocities in a cosmology in which dark matter interacts with baryons.
With this prescription, we derive CMB bounds on the cross section, excluding
$\sigma_0 > 2.3 \times 10^{-33}~\mathrm{cm}^2$ for $n=-2$ and $\sigma_0 > 1.7
\times 10^{-41}~\mathrm{cm}^2$ for $n=-4$ at $95\%$ confidence, for dark matter
masses below 10 MeV. Furthermore, we investigate how these constraints change
when only a subcomponent of dark matter is interacting. We show that Planck
limits vanish if $\lesssim 0.4\%$ of dark matter is tightly coupled to baryons.
We discuss the implications of our results for present and future cosmological
observations.
