Consider a system consisting of a linear wave equation coupled to
a transport equation:
\begin{equation*}
\Box_{t,x}u =f ,
\end{equation*}
\begin{equation*}
(\partial_t + v(\xi) \cdot \nabla_x)f =P(t,x,\xi, D_\xi)g ,
\end{equation*}
Such a system is called \textit{nonresonant} when the maximum speed
for particles governed by the transport equation is less than the
propagation speed in the wave equation. Velocity averages of
solutions to such nonresonant coupled systems are shown to be more
regular than those of either the wave or the transport equation
alone. This smoothing mechanism is reminiscent of the proof of
existence and uniqueness of $C^1$ solutions of the Vlasov-Maxwell
system by R. Glassey and W. Strauss for time intervals on which
particle momenta remain uniformly bounded, see
``Singularity formation in a collisionless plasma could occur only
at high velocities'', \textit{Arch. Rational Mech. Anal.}
\textbf{92} (1986), no. 1, 59-90. Applications of our smoothing
results to solutions of the Vlasov-Maxwell system are discussed.