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The observational value of the velocity dispersion, $Δ\upsilon$, 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 $Δ\upsilon$ 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 $10^{-3} eV < m_χ< 10^9 eV$, finding $Δ\upsilon_0 \lesssim$ 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.