  We forecast constraints on dark matter (DM) scattering with baryons in the
early Universe with upcoming and future cosmic microwave background (CMB)
experiments, for DM particle masses down to 15 keV. In terms of the upper limit
on the interaction cross section for a velocity-independent spin-independent
elastic scattering, compared to current Planck results, we find a factor of
$\sim$6 improvement with CMB-Stage 3, a factor of $\sim$26 with CMB-Stage 4,
and a factor of $\sim$200 with a cosmic-variance limited experiment. Once the
instrumental noise reaches the proximity of 1 $\mu$K-arcmin, the constraints
are entirely driven by the lensing measurements. The constraints benefit from a
wide survey, and show gradual improvement for instrumental noise levels from 10
$\mu$K-arcmin to 1 $\mu$K-arcmin and resolution from 5 arcmin to 1 arcmin. We
further study degeneracies between DM interactions and various other signatures
of new physics targeted by the CMB experiments. In the primary temperature and
polarization only, we find moderate degeneracy between the effects of DM
scattering, signals from massive neutrinos, and from the effective number of
relativistic degrees of freedom. The degeneracy is almost entirely broken once
the lensing convergence spectrum is included into the analyses. We discuss the
implications of our findings in context of planned and upcoming CMB
measurements and other cosmological probes of dark-sector and neutrino physics.
