  We adapt the vector field method of Klainerman to the study of relativistic
transport equations. First, we prove robust decay estimates for velocity
averages of solutions to the relativistic massive and massless transport
equations, without any compact support requirements (in $x$ or $v$) for the
distribution functions. In the second part of this article, we apply our method
to the study of the massive and massless Vlasov-Nordstr\"om systems. In the
massive case, we prove global existence and (almost) optimal decay estimates
for solutions in dimensions $n \geq 4$ under some smallness assumptions. In the
massless case, the system decouples and we prove optimal decay estimates for
the solutions in dimensions $n \geq 4$ for arbitrarily large data, and in
dimension $3$ under some smallness assumptions, exploiting a certain form of
the null condition satisfied by the equations. The $3$-dimensional massive case
requires an extension of our method and will be treated in future work.
