  We develop a new approach to velocity averaging lemmas based on the
dispersive properties of the kinetic transport operator. This method yields
unprecedented sharp results, which display, in some cases, a gain of one full
derivative. Moreover, the study of dispersion allows to treat the case of
$L_x^rL^p_v$ integrability with $r\leq p$. We also establish results on the
control of concentrations in the degenerate $L_{x,v}^1$ case, which is
fundamental in the study of the hydrodynamic limit of the Boltzmann equation.
