Weak convergence of additive functionals of a sequence of Markov chains

We consider additive functionals $\phi_n$, $n\geq1$ defined on a sequence of Markov chains that weakly converges to a Markov process. We give sufficient condition for $\phi_n$, $n\geq1$ to converge in distribution, formulated in the terms of their characteristics (i.e. expectations). This condition generalizes Dynkin's theorem on convergence of $W$-functionals of a time homogeneous Markov process.