When does convergence of a sequence of stopped processes with independent increments imply convergence of the non-stopped processes

We give two criterions which show when convergence in law of a sequence of processes with independent increments, stopped at their first jump within given size, implies convergence of the non-stopped processes; if this result can appear to fail, it is always true for instance when the limiting process has no fixed time of discontinuity. As an application, we give settings where convergence of the processes stopped a short while after this first time of 'big' jump implies convergence of the non-stopped processes.