Stroock's Markov processes book is, as far as I know, the most readily accessible treatment of inhomogeneous Markov processes: he does all the basics in the context of simulated annealing, which is neat. Kleinrock's volume 1 is also of interest, though "buggy" IIRC.

In my experience the key object is the propagator $U(t) := \mathcal{TO}^* \int_0^t Q(s) \ ds$, where $Q$ is the time-dependent generator and $\mathcal{TO}^*$ is the formal adjoint or reverse time-ordering operator (see here and here for forward time-ordering). With this in hand, the transition kernel can be expressed as $P(s,t) = U^{-1}(s)U(t)$. Thus, e.g., an initial distribution $p(0)$ is propagated as $p(t) = p(0)U(t)$, which gives you the essential stuff to get a handle on (e.g.) mixing times.

One way to guarantee convergence is to have $U(t)$ varying within the group fixing a distribution, cf. Has the Lie group preserving a probability distribution been used in Bayesian statistics?

As another push towards your goal, the Dynkin martingale formula becomes (under suitable conditions, e.g., $t \mapsto f_t$ is $C^1$ and $f_t(X_t)$ is bounded)
$$\mathbb{E}(f_t(X_t)-f_0(X_0)) = \mathbb{E}\int_0^t(\partial_s+Q(s))f_s \circ X_s \ ds$$
Here Rogers and Williams IV.20-21 is of interest, since the extension to inhomogeneous processes is trivial.

Finally, I will note that the cutest time-inhomogenous Markov process is the Poisson process, cf. http://blog.eqnets.com/2009/07/28/why-poissonian-traffic-models-matter-more-now-than-ever-part-4/