  The central limit theorem is one of the most fundamental results in
probability and has been successfully extended to locally dependent data and
strongly-mixing random fields. In this paper, we establish its rate of
convergence for transport distances, namely for arbitrary $p\ge1$ we obtain an
upper bound for the Wasserstein-$p$ distance for locally dependent random
variables and strongly mixing stationary random fields. Our proofs adapt the
Stein dependency neighborhood method to the Wasserstein-$p$ distance and as a
by-product we establish high-order local expansions of the Stein equation for
dependent random variables. Finally, we demonstrate how our results can be used
to obtain tail bounds that are asymptotically tight, and decrease polynomially
fast, for the empirical average of weakly dependent random variables.
