  This paper is devoted to the study of a certain type of martingale problems
associated to general operators corresponding to processes which have finite
lifetime. We analyse several properties and in particular the weak convergence
of sequences of solutions for an appropriate Skorokhod topology setting. We
point out the Feller-type features of the associated solutions to this type of
martingale problem. Then localisation theorems for well-posed martingale
problems or for corresponding generators are proved.
