  This study investigates the $L^1_{\operatorname{loc}}$ compactness of
velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic
equations. The equations are examined within both deterministic and stochastic
heterogeneous environments. The primary objective is to deduce velocity
averaging results under conditions on $u_n$ and the drift ${\mathfrak
f}={\mathfrak f}(t,{\boldsymbol x},{\boldsymbol \lambda})$ that are more
lenient than those stipulated in previous studies. The main outcome permits the
inclusion of highly irregular drift vectors ${\mathfrak f} \in L^q$ that adhere
to a general non-degeneracy condition. Moreover, the sequence $\{u_n\}$ is
uniformly bounded in $L^p$ -- for an exponent $p$ allowed to be strictly
smaller than $2$ -- under the requirement $\frac{1}{p} + \frac{1}{q} < 1$.
Resolving the matter of strong compactness in velocity averages, considering
these assumptions, has remained an open problem for a long time. The
cornerstone of our work's progress lies in the strategic employment of the
broader concept of $H$-distributions, moving beyond the traditional reliance on
$H$-measures. Notably, our study represents one of the first significant uses
of $H$-distributions in this context.
