Background: The sequence of Binomial distributions $\text{Bin}(n,p_n)$ with parameters $(n,p_n)$ is known to converge as $n\rightarrow \infty$ to the Poisson distribution $\text{Poi}(\lambda)$ if $p_n\cdot n\rightarrow \lambda$. A special case of this is given by the following choice $p_n=\lambda/n$.
I would like to claim that the same convergence holds when I move to the expected values (under $\text{Bin}(n,p_n)$ and $\text{Poi}(\lambda)$) for a given function $f:\mathbb{N}_0\rightarrow \mathbb{R}_{\ge0}$.
Setup and question: Given a function $f:\mathbb{N}_0\rightarrow \mathbb{R}_{\ge0}$, non-negative, non-decreasing and convex (in the discrete sense) over the domain $\mathbb{N}_0$, I would like to show that the following holds
\begin{equation} \lim_{n\rightarrow\infty}\mathbb{E}_{X\sim\text{Bin}(n,\lambda/n)}[f]= \mathbb{E}_{X\sim\text{Poi}(\lambda)}[f], \end{equation}
i.e. that I can 'interchange' the expected value and the limiting operation and conclude.
What I have tried:
What makes this challenging is the fact that $f$ is not bounded. For example, I would like this result to hold for polynomials, and beyond.
I think it should be easy to show that if the left hand side is unbounded, so is the RHS, so we have equality in this case.
Hence, we can restrict to the case where we assume $\mathbb{E}_{X\sim\text{Poi}(\lambda)}[f]<\infty$. I have tried pursuing different directions e.g. Tannery's theorem, dominated/monotone convergence, even Fatou's Lemma, though I do not seem to be able to find the "right combination" or idea.
Thank you!
Edit: a partial answer is available when one cares only about the moments, i.e. for the special choice of $f(k)=k^d$ here.
I need a more general answer that applies beyond this case.