  We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$
under Gaussian smoothing. It is well known that the heat semigroup is
exponentially contractive with respect to the Wasserstein distances on
manifolds of positive curvature; however, on flat Euclidean space---where the
heat semigroup corresponds to smoothing the measures by Gaussian
convolution---the situation is more subtle. We prove precise asymptotics for
the $2$-Wasserstein distance under the action of the Euclidean heat semigroup,
and show that, in contrast to the positively curved case, the contraction rate
is always polynomial, with exponent depending on the moment sequences of the
measures. We establish similar results for the $p$-Wasserstein distances for $p
\neq 2$ as well as the $\chi^2$ divergence, relative entropy, and total
variation distance. Together, these results establish the central role of
moment matching arguments in the analysis of measures smoothed by Gaussian
convolution.
