  We show the $L^2$-Wasserstein contraction for the transition kernel of a
discretised diffusion process, under a contractivity at infinity condition on
the drift and a sufficiently high diffusivity requirement. This extends recent
results that, under similar assumptions on the drift but without the
diffusivity restrictions, showed the $L^1$-Wasserstein contraction, or
$L^p$-Wasserstein bounds for $p > 1$ that were, however, not true contractions.
We explain how showing the true $L^2$-Wasserstein contraction is crucial for
obtaining the local Poincar\'{e} inequality for the transition kernel of the
Euler scheme of a diffusion. Moreover, we discuss other consequences of our
contraction results, such as concentration inequalities and convergence rates
in KL-divergence and total variation. We also study the corresponding
$L^2$-Wasserstein contraction for discretisations of interacting diffusions. As
a particular application, this allows us to analyse the behaviour of particle
systems that can be used to approximate a class of McKean-Vlasov SDEs that were
recently studied in the mean-field optimization literature.
