arXiv:2511.08111v3 Announce Type: replace-cross Abstract: We develop an operator-theoretic framework for the contraction properties of Markov semigroups with respect to a general class of Kantorovich semi-distances, which includes Wasserstein distances and weighted total variation norms. The framework is based on Kantorovich contraction coefficients, which extend Dobrushin's ergodic coefficient from the total variation distance to general transport costs. Our main result shows that a geometric drift condition, combined with a local contraction condition on the sub-level sets of the Lyapunov function, implies that the Markov operator is a strict contraction with respect to an explicit modification of the underlying Kantorovich cost. This yields, in a unified way, the existence and uniqueness of invariant measures and exponential decay estimates in Wasserstein distances, for both discrete time and continuous time Markov semigroups, as well as for time-varying semigroups. We illustrate the scope of the framework with (i) Markov transitions on models with boundary states, including bounded domains with entrance boundaries, (ii) operator products of a Markov kernel and its adjoint, including two-block Gibbs samplers, (iii) iterated random functions, including Euler-type schemes, (iv) diffusion models, including overdamped Langevin diffusions with potentials that are convex at infinity, jump-diffusions on compact manifolds, and Jacobi and Riccati matrix diffusions, and (v) time-varying and nonlinear semigroups, including Feynman-Kac distribution flows and Sinkhorn bridges. In several of these settings we obtain new contraction estimates, while in others we recover known results through short and elementary arguments.