Asymptotically ergodic Markov functionals of an ergodic process

Let $X(t)$ be a homogeneous Markov process given on a state space $(E,\mathcal{B})$ and having an invariant distribution $\pi ( \cdot )$. Let $\{ \xi _n (t)\} $ be a sequence of cut-off Markov functionals with killing times $\{ \xi _n \} $ and a set of values $I = \{ 1,2, \ldots ,d\} $ which converges to a trivial functional with a stationary distribution $\rho ( \cdot )$. We give the conditions under which there exists a sequence $\varepsilon _n \to + 0$ such that if the inequality $\mathbf{P}_{\pi ,\rho } \{ \xi _n < \infty \} > 0$ holds for all sufficiently large $n$, then for any $t \ge 0,x \in E$, $i,j \in I$, and all continuous bounded functions $\varphi (y),y \in E$,
$$ \lim_{n\to\infty}\mathbf{P}_{x,i}\biggl[\varphi\biggl(X\biggl(\frac t {\varepsilon_n}\biggr)\biggr),\varepsilon_n\biggl(\frac t{\varepsilon_n}\biggr)=j\biggr]=e^{-t}\rho(j)\int_E\pi(dy)\varphi(y) $$