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 Savaré 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 Poincaré 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 Rényi 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.