For each natural number $N \geq 1$, let $X^N = (X^N_k)_{k = 1}^{\infty}$ be an homogeneous Markov process taking values in $R_+$, with transition kernel \begin{equation} p_N(A|X^{N}_{k-1}) = P[X^{N}_{k} \in A|X^{N}_{k-1}], \quad A \in B(R_+), \end{equation} and let $X = (X_k)_{k = 1}^{\infty}$ be another homogeneous Markov process with transition kernel \begin{equation} p(A|X_{k-1}) = P[X_k \in A|X_{k-1}], \quad A \in B(R_+). \end{equation} Say all the Markov process defined above share the same initial distribution $\pi$.
If $p_N(\cdot|x) \to p(\cdot|x)$ (weakly), for every $x \in R_+$. Does it always hold that $X^N \to X$ in distribution?
If the answer is yes, how can it be proved?
If the answer is no, are there any aditional assumptions under which the statement holds?