Suppose $(X_t)_{t \geq 0}$ is a Markov chain on the state space $S$ with transition probability $p$, and that $\pi$ is a stationary distribution for $p$. If $X_0 \sim \pi$, then we know $X_t \sim \pi$ for all $t \geq 0$. However, if $T$ is a stopping time, then it might not be the case that $X_T \sim \pi$. For instance, if our Markov chain is simple random walk on $\mathbb{Z}_n$, then $\pi(x)=\frac{1}{n}$ is a stationary distribution; yet if $X_0 \sim \pi$ and $T=\inf\{t \geq 0 \mid X_t=0\}$, then $X_T = 0$, so $X_T \not\sim\pi$.
My question is whether there are any conditions on the Markov chain, stationary distribution, and/or stopping time which allow you to conclude that if $X_0 \sim \pi$ then $X_T \sim \pi$.