This talk deals with quantifying the rate of contraction of the relative entropy with respect to a reference probability measure under the action of a Markov kernel. Sharp contraction estimates for Markov kernels are useful in a variety of settings, including distributed communication and simulation, exact and approximate stochastic filtering, analysis of MCMC algorithms, and statistical physics. I will present a method for obtaining such estimates that relies on information-transportation inequalities, which were introduced by Katalin Marton in a series of breakthrough papers. I will illustrate this method through examples of random walks on graphs and information flow between disjoint subsets of spins in probabilistic graphical models.