  We study the relationship between functional inequalities for a Markov kernel
on a metric space $X$ and inequalities of transportation distances on the space
of probability measures $\mathcal{P}(X)$. Extending results of Luise and
Savar\'e on Hellinger--Kantorovich contraction inequalities for the particular
case of the heat semigroup on an $RCD(K,\infty)$ metric space, we show that
more generally, such contraction inequalities are equivalent to reverse
Poincar\'e inequalities. We also adapt the "dynamic dual" formulation of the
Hellinger--Kantorovich distance to define a new family of divergences on
$\mathcal{P}(X)$ which generalize the R\'enyi divergence, and we show that
contraction inequalities for these divergences are equivalent to the reverse
logarithmic Sobolev and Wang Harnack inequalities. We discuss applications
including results on the convergence of Markov processes to equilibrium, and on
quasi-invariance of heat kernel measures in finite and infinite-dimensional
groups.
