Let us assume that we are given a time inhomogenous Markov chain in continuous time (ICTMC) $(X(t))_{t \geq0}$ with a finite state space $\{1,\ldots,n\}$. Set $P(t)_{i,j} := \mathbb{P}(X(t) = j \mid X(0) = i)$ and let $Q(t)$ denote the time dependent transition rates. Further, define $\pi_i(t) := \mathbb{P}(X(t) = i)$ and fix an initial probability distribution $\pi(0)$.
After skimming through the very few books that I was able to find on the topic of ICTMCs, I discovered that the Chapman-Kolmogorov equations $\dot{\pi}(t) = \pi(t) Q(t)$ carry over to ICTMC. However, the corresponding book [1] makes the assumption that $t \mapsto q_{i,j}(t)$ is continuous for all $1 \leq i,j \leq n$. I expect the result to hold true if $t \mapsto q_{i,j}(t)$ are piecewise-continuous (or, if it helps, piecewise constant), but was not able to find any reference that covers this case.
Does anybody know whether this holds true? Any pointer to a reference that could discuss this would be more than welcome.
[1] Gikhman, Skorohod - Introduction to the Theory of Random Processes