For $k \in \mathbb{N}$ let $X^k = (X^k_n)_{n \in \mathbb{N}}$ and $X = (X_n)_{n \in \mathbb{N}}$ be Markov chains on a Polish space $E$ with initial distributions $\alpha^k$ and $\alpha$ respectively and transition probability kernels $P^k$ and $P$ respectively. By Karr's theorem, if $\alpha^k \Rightarrow \alpha$ (weak convergence, i.e. with respect to $C_b(E)$) and $P^k(x^k, \cdot) \Rightarrow P(x, \cdot)$ whenever $x^k \to x$ then $X^k$ converges in distribution to $X$.
The distribution of a Markov chain $X$ with random lifetime (possibly finite) can be described by an initial subdistribution and a transition subprobability kernel $P$. As a process, this Markov chain $X$ is considered as a Markov chain with infinite lifetime on $E \cup \{ \Delta \}$ for some coffin state $\Delta$. If $E$ is locally compact then one can equip $E \cup \{ \Delta \}$ with the one-point compactification topology. Does Karr's theorem extend to this setup? In other words, if $X^k$ and $X$ are Markov chains with random lifetime, initial subdistributions $\alpha^k$ and $\alpha$ and transition subprobability kernels $P^k$ and $P$ such that $\alpha^k \to \alpha$ (in some appropriate sense, I think this should then be vague convergence, i.e. with respect to $C_0(E)$) and $P^k(x^k, \cdot) \to P(x, \cdot)$ (also vague convergence) whenever $x^k \to x$ then $X^k$ converges in distribution to $X$ as processes in $E \cup \{ \Delta \}$.