  If dark matter has even been in sufficient thermal contact with the visible
sector and sufficiently light ($m_\chi\lesssim\mathcal{O}(10)~\text{keV}$), the
thermal motion inherited from the visible sector will cause significant free
streaming effect which is subject to the structure formation constraint,
similar to the benchmark thermal warm dark matter model. Here we identify the
interaction responsible for such thermal contact to be the interaction probed
by the deep underground dark matter direct detection experiments. With the
kinetic decoupling technique on the $m_\chi$ vs. $\sigma$ plot we determine the
bound shape in detail, and find that recasting the current Lyman-$\alpha$ bound
gives a constraint of $m_\chi\gtrsim73~\text{keV}$, and it gets relaxed to
$m_\chi\gtrsim35~\text{keV}$ for a smaller cross section of
$\sigma<10^{-46}~\text{cm}^2$ with some model dependence. That can be taken as
a generic ``no go'' constraint for light dark matter direct detection
experiments, and the known caveats are if dark matter is axion-like with an
early Bose-Einstein condensation form, or if there is Brownian motion
protection of the free streaming.
