Summary: The basic goal is to find the time evolution of the probability generating function (or the moment generating function or the characteristic function if you prefer) for a continuous-time markov chain with a finite number of states.
To set notation for a particular time period: let $X$ be a random variable with a discrete number of states $n = 1,\ldots N$. Let the probability distribution of this be $\mathbb{P}_n \equiv Pr(X = n)$, with the usual properties such as $\sum_{n=1}^N \mathbb{P}_n = 1$. Furthermore, use the standard definition of the probability generating function of this random variable as $$ G(z) \equiv \sum_{n=1}^{N}\mathbb{P}_n z^n $$ (Note that if we summed to infinity, it would be equivalent as it only has support up to $N$). Finally, summarize the probability distribution by the vector $\mathbb{P} \equiv \{\mathbb{P}_1, \mathbb{P}_2,\ldots \mathbb{P}_N\}\in\mathbb{R}^N$. Note that typically the conversion between $\mathbb{P}$ and $G(z)$ is done with a z-transform.
Stochastic process for $X$: In reality, the random variable $X$ evolves as a continuous time stochastic process $X_t$ for $t\geq 0$. The process is a continuous-time markov chain (http://en.wikipedia.org/wiki/Continuous-time_Markov_chain) with some infinitesimal generator $\mathbb{Q} \in \mathbb{R}^{N\times N}$
Denote $\mathbb{P}_n(t) \equiv Pr(X_t = n)$, stacking the vector of probabilities at any particular point in time as $\mathbb{P}(t)$. With this, given some initial condition of the probability distribution $\mathbb{P}(0)$, the evolution of the probability distribution is given by solving the Kolmogorov Forward Equation as a system of $N$ ODEs, $$ \frac{d\mathbb{P}(t)}{d t} = \mathbb{P}(t)\cdot \mathbb{Q},\,\text{subject to }\mathbb{P}(0) $$ Defining with matrix exponentials, this ODE has the general solution of $$\mathbb{P}(t) = e^{t\mathbb{Q}}$$
Of course, I could denote the probability generating function of $X_t$ as $$ G(t, z) \equiv \sum_{n=1}^{\infty}\mathbb{P}_n(t) z^n $$
Just to be clear: The typical properties of the probability generating function, $G(t,z)$, only apply to the non-time dimensions, i.e. $z$.
Up until now, everything is completely standard and follows the usual definitions. My basic question is whether we can write an expression for the evolution of $G(t,z)$ either directly, or as a partial differential equation that I could solve numerically. Why would this be useful? Because $\frac{d^K G(t,1)}{ d z^K}$ are the factorial moments of the random variable $X_t$ for a fixed time $t$. See http://en.wikipedia.org/wiki/Probability-generating_function#Probabilities_and_expectations
Version 1 of Question: From $\mathbb{Q}$ is there a way to write a differential equation for $G(t,z)$ directly?
Version 2 of Question: is there any clean way to write/simplify $G(t,z)$ in terms of $e^{t \mathbb{Q}}$ (other than the obvious, direct unsimplified substitution into the definition)? Feel free to use an eigen-decomposition $\mathbb{Q} \equiv U D U^{-1}$ so that $e^{t \mathbb{Q}} = U e^{t D}U^{-1}$ for some diagonal $D$
As a simple example: the telegraph process with switching intensity $\lambda > 0$ has $N=2$ and infinitesimal generator $\mathbb{Q} = \begin{bmatrix}-\lambda & \lambda\\ \lambda & - \lambda\end{bmatrix} $. If we started in state $1$ with certainty, i.e., $\mathbb{P}_1(0) = 1$, then with a $G(t,z)$ for this $\mathbb{Q}$, we could find the evolution of the moments over time by taking the derivatives of $G(t,z)$ at $z=1$. Also, see http://www.columbia.edu/~ww2040/TransientMM1questa.pdf for an example of this with a particular stochastic process (albeit one with an infinite support for $n$).
One last point: if you prefer to work with the characteristic function, the moment generating function, or anything else with an equivalence to the the probability distribution function, that is fine.
