Tightness for processes with fixed points of discontinuities and applications in varying environment

We establish a sufficient condition for the tightness of a sequence of stochastic processes. Our condition makes it possible to study processes with accumulations of fixed times of discontinuity. 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.