Let $\{X(t)\}_{t\geq 0}$ denote an irreducible continuous-time Markov chain with finite state space and unknown generator matrix. Let $P_G(X(s)=j|X(0)=k,X(t)=l)$ denote the conditional probability of the CTMC with generator $G$ being in state $j$ at time $s$, given that it is in state $k$ at time $0$ and state $l$ at time $t$. I want to show that the matrix of such probabilities with $k,l$th element $P_G(X(s)=j|X(0)=k,X(t)=l)$ is Lipschitz in $G$ for Frobenius norm in a small neighborhood of $G^*$.
One way to do that is to use Bayes theorem to compute the entries of each matrix, giving us $\frac{P_G(X(t)=l|X(s))P(X(s)|X(0)=k)}{P_G(X(t)=l|X(0)=k)}$. However, the need to deal with $P_G(X(t)=l|X(0)=k)$ (assuming that it has a lower bound in a ball around $G^*$) leads to constants that aren't as small as I would hope for. Is there some way to show that it is Lipschitz without applying Bayes theorem?