For a discrete-time, nonhomogeneous Markov process, how would you proceed to compute $Pn(t)$ from a given transition matrix $T(t)$?
For homogeneous chains there are two alternative approaches. You either diagonalize $T$, or write the differential equation for the generating function. But for inhomogeneous chains? Diagonalizing won't help, because the transition matrix changes in time, and generating functions meet with obvious complications. So I wonder how you would proceed.
Also, do you know of any case in the literature where explicit formulas have been obtained for $P_n(t)$ in discrete, inhomogeneous Markov chain with more than two states? I'm sure there are many, I'm just looking for examples of the techniques that have been used.
Instead of starting with the Markov chain and finding the probabilities $P_n(t)$, consider starting with the probabilities and finding a Markov chain that gives those probabilities.
For a sequence of invertible right stochastic matrices $Q(t)$ with $Q(0) = I$ (where $Q_{ij}(t)$ is to represent the probability of going from state $i$ at time $0$ to state $j$ at time $t$), take the transition matrices $P(t) = Q(t)^{-1} Q(t+1)$. As long as all entries are nonnegative, you have an example.
One way to do this is to start with a homogeneous continuous-time Markov chain $\tilde{Q}(t)$ and let $Q(m) = \tilde{Q}(t_m)$ for some increasing sequence of times $t_m$.