In some central models in life insurance mathematics, the state of the insured is modeled using a continuous-time time-inhomogeneous Markov process with finitely many states. While many results for such processes can be argued for heuristically by analogies with the homogeneous case, I am finding it difficult to obtain rigorous reference books or articles on such processes.
What I am looking for is essentially something akin to Norris' "Markov chains" or Brémaud's "Markov chains: Gibbs fields, Monte Carlo simulation and queues", but for Markov processes with time-dependent intensities.
As this class of Markov processes is a special case of marked point processes with finite state space, I would also consider references which generally consider MMP's, but which preferably also specialize their results to the case of Markov processes.