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We study a Markov process with two components: the first component evolvesaccording to one of finitely many underlying Markovian dynamics, with a choiceof dynamics that changes at the jump times of the second component. The secondcomponent is discrete and its jump rates may depend on the position of thewhole process. Under regularity assumptions on the jump rates and Wassersteincontraction conditions for the underlying dynamics, we provide a concretecriterion for the convergence to equilibrium in terms of Wasserstein distance.The proof is based on a coupling argument and a weak form of the Harristheorem. In particular, we obtain exponential ergodicity in situations which donot verify any hypoellipticity assumption, but are not uniformly contractingeither. We also obtain a bound in total variation distance under a suitableregularising assumption. Some examples are given to illustrate our result,including a class of piecewise deterministic Markov processes.

Exponential ergodicity for Markov processes with random switching