  We consider $\mathbb{R}^d$-valued diffusion processes of type
  \begin{align*}
  dX_t\ =\ b(X_t)dt\, +\, dB_t.
  \end{align*} Assuming a geometric drift condition, we establish contractions
of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with
explicit constants. Our results are in the spirit of Hairer and Mattingly's
extension of Harris' Theorem. In particular, they do not rely on a small set
condition. Instead we combine Lyapunov functions with reflection coupling and
concave distance functions. We retrieve constants that are explicit in
parameters which can be computed with little effort from one-sided Lipschitz
conditions for the drift coefficient and the growth of a chosen Lyapunov
function. Consequences include exponential convergence in weighted total
variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich
contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently
weak but not necessarily bounded nonlinearities. We also establish quantitative
bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.
