Let $\{X_n(t) \in \mathbb{R}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique stationary distribution, $X_{\infty}(t) \in \mathbb{R}^+$, i.e., $$X_n(t) \rightarrow^w X_{\infty}(t). $$
Suppose for each $n$, $X_n(t)$ converges weakly to $\overline{X}_n$ as $t \rightarrow 0$, i.e., $$X_n(t) \rightarrow^w \overline{X}_n$$
The sequence $\{\overline{X}_n \in \mathbb{R}^+\}$ is also a Markov chain. Suppose it has a unique stationary distribution $\overline{X}_{\infty} \in \mathbb{R}^+$, i.e., $$\overline{X}_n \rightarrow^w \overline{X}_\infty.$$
Are there general conditions under which $X_{\infty}(t)$ converge to $\overline{X}_{\infty}$ as $t \rightarrow \infty$, i.e., $$ X_{\infty}(t) \rightarrow^w \overline{X}_{\infty}.$$
I believe that the above problem would require some justification for an exchange of limits. I want to know if under Markovian conditions there are some results which justify these results.