  The observational value of the velocity dispersion, $\Delta\upsilon$, is
missing in the Dark Matter (DM) puzzle. Non-zero or non-thermal DM velocities
can drastically influence Large Scale Structure and the 21-cm temperature at
the epoch of the Cosmic Dawn, as well as the estimation of DM physical
parameters, such as the mass and the interaction couplings. To study the
phenomenology of $\Delta\upsilon$ we model the evolution of DM in terms of a
simplistic and generic Boltzmann-like momentum distribution. Using cosmological
data from the Cosmic Microwave Background, Baryonic Acoustic Oscillations, and
Red Luminous Galaxies, we constrain the DM velocity dispersion for a broad
range of masses $10^{-3} eV < m_\chi < 10^9 eV$, finding $\Delta\upsilon_0
\lesssim$ 0.33 km/s (99% CL). Including the EDGES $T_{21}$-measurements, we
extend our study to constrain the baryon-DM interaction in the range of DM
velocities allowed by our analysis. As a consequence, we present new bounds on
two electromagnetic models of DM, namely minicharged particles (MCPs) and
electric dipole moment (EDM). For MCPs, the parameter region that is consistent
with EDGES and independent bounds on cosmological and stellar physics is very
small, pointing to the sub-eV mass regime of DM. A window in the MeV-GeV may
still be compatible with these bounds for MCP models without a hidden photon.
But the EDM parameter region consistent with EDGES is excluded by Big-Bang
Nucleosynthesis and Collider Physics.
