Dynamic Systems – Is There a Foster-Lyapunov Criterion for Time-varying Markov Processes?

Suppose I have two Markov processes with the transition kernels Q_1 (y | x) and Q_2 (y | x). Suppose I also have Lyapunov functions V_1, V_2 for these processes w.r.t. there is a compact set S, so that for x in S the drift of the respective Lyapunov function is bounded below by a negative number for both processes.

Suppose I construct a new Markov process by using the kernel Q_1 at odd times and the kernel Q_2 at even times.
Can I guarantee the stability of this new process in the set S, d. H. Can I construct a Lyapunov function for this third process in set S?