A continuous time homogeneous Markov chain $X_t$ over a finite state space $\{ 1, \dots, n \}$ satisfies the property
$$P(X(s+t) = j \mid X(t) = i) = P(X(s) = j \mid X(0) = i).$$
If $S$ and $T$ are random times is it then true that
$$P(X(S+T) = j \mid X(T) = i) = P(X(S) = j \mid X(0) = i).$$
Edit: See the comments for a nice counterexample in the case $S$ deterministic and $T$ random but not stopping time (for discrete time Markov chains). Now what happens if we restrict $T$ to be a stopping time?