  We here study random evolutions on Banach spaces, driven by a class of
semi-Markov processes. The expectation (in the sense of Bochner) of such
evolutions is shown to solve some abstract Cauchy problems. Further, the
abstract telegraph (damped wave) equation is generalized to the case of
semi-Markov perturbations. A special attention is devoted to semi-Markov models
of scattering transport processes which can be represented through these
evolutions. In particular, we consider random flights with infinite mean flight
times which turn out to be governed by a semi-Markov generalization of a linear
Boltzmann equation; their scaling limit is proved to converge to superdiffusive
transport processes.
