  We prove precise stability results for overshoots of Markov additive
processes (MAPs) with finite modulating space. Our approach is based on the
Markovian nature of overshoots of MAPs whose mixing and ergodic properties are
investigated in terms of the characteristics of the MAP. On our way we extend
fluctuation theory of MAPs, contributing among others to the understanding of
the Wiener-Hopf factorization for MAPs by generalizing Vigon's \'equations
amicales invers\'es known for L\'evy processes. Using the Lamperti
transformation the results can be applied to self-similar Markov processes.
Among many possible applications, we study the mixing behavior of stable
processes sampled at first hitting times as a concrete example.
