  We study a Markov process with two components: the first component evolves
according to one of finitely many underlying Markovian dynamics, with a choice
of dynamics that changes at the jump times of the second component. The second
component is discrete and its jump rates may depend on the position of the
whole process. Under regularity assumptions on the jump rates and Wasserstein
contraction conditions for the underlying dynamics, we provide a concrete
criterion for the convergence to equilibrium in terms of Wasserstein distance.
The proof is based on a coupling argument and a weak form of the Harris
theorem. In particular, we obtain exponential ergodicity in situations which do
not verify any hypoellipticity assumption, but are not uniformly contracting
either. We also obtain a bound in total variation distance under a suitable
regularising assumption. Some examples are given to illustrate our result,
including a class of piecewise deterministic Markov processes.
