  We investigate the problem of quantifying contraction coefficients of Markov
transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion
processes, relatively precise quantitative bounds on contraction rates have
recently been derived by combining appropriate couplings with carefully
designed Kantorovich distances. In this paper, we partially carry over this
approach from diffusions to Markov chains. We derive quantitative lower bounds
on contraction rates for Markov chains on general state spaces that are
powerful if the dynamics is dominated by small local moves. For Markov chains
on $\mathbb{R^d}$ with isotropic transition kernels, the general bounds can be
used efficiently together with a coupling that combines maximal and reflection
coupling. The results are applied to Euler discretizations of stochastic
differential equations with non-globally contractive drifts, and to the
Metropolis adjusted Langevin algorithm for sampling from a class of probability
measures on high dimensional state spaces that are not globally log-concave.
