A sufficient condition for the tightness of time-inhomogeneous Markov processes

We establish a sufficient condition for the tightness of a sequence of time-inhomogeneous strong Markov processes. Our condition makes it possible to study processes with accumulations of fixed times of discontinuity, a case where Aldous' criterion fails. Our motivation comes from the study of processes in varying or random environment. We demonstrate the usefulness of our condition on two examples: Galton Watson branching processes in varying environment and logistic branching processes with catastrophes.