  We consider elliptic diffusion processes on $\mathbb R^d$. Assuming that the
drift contracts distances outside a compact set, we prove that, at a
sufficiently high temperature, the Markov semi-group associated to the process
is a contraction of the $\mathcal W_2$ Wasserstein distance, which implies a
Poincar\'e inequality for its invariant measure. The result doesn't require
neither reversibility nor an explicit expression of the invariant measure, and
the estimates have a sharp dependency on the dimension. Some variations of the
arguments are then used to study, first, the stability of the invariant measure
of the process with respect to its drift and, second, systems of interacting
particles, yielding a criterion for dimension-free Poincar\'e inequalities and
quantitative long-time convergence for non-linear McKean-Vlasov type processes.
