  This work is concerned with the broad question of propagation of regularity
for smooth solutions to non-linear Vlasov equations. For a class of equations
(that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), we prove that
higher regularity in space is propagated, locally in time, into higher
regularity for the moments in velocity of the solution. This in turn can be
translated into some anisotropic Sobolev higher regularity for the solution
itself, which can be interpreted as a kind of weak propagation of space
regularity. To this end, we adapt the methods introduced in the context of the
quasineutral limit of the Vlasov-Poisson system in [D. Han-Kwan and F. Rousset,
Ann. Sci. \'Ecole Norm. Sup., 2016].
