  We extend the celebrated Stone's theorem to the framework of distributional
regression. More precisely, we prove that weighted empirical distribution with
local probability weights satisfying the conditions of Stone's theorem provide
universally consistent estimates of the conditional distributions, where the
error is measured by the Wasserstein distance of order p $\ge$ 1. Furthermore,
for p = 1, we determine the minimax rates of convergence on specific classes of
distributions. We finally provide some applications of these results, including
the estimation of conditional tail expectation or probability weighted moment.
