Let $( \mathbb{P}_t )_{ t \geq 0 }$ be a family of probability measures on the measurable space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ such that $$ \int_{\mathbb{R}} x^2 \mathbb{P}_t (dx) < \infty \quad \forall t \geq 0, $$ and assume that the mapping $$ [0, \infty) \ni t \mapsto \mathbb{P}_t $$ is continuous with respect to the $2$nd Wasserstein metric denoted by $W_2(\cdot, \cdot)$.
Now let $f : \mathbb{R} \times [0, \infty) \rightarrow [0, \infty]$ be a measurable function.
Is it true that the mapping $$ [0, \infty) \ni t \mapsto \int_{\mathbb{R}} f(x,t) \mathbb{P}_t(dx) \tag{1} $$ is measurable, i.e., $\mathcal{B}([0, \infty))-\mathcal{B}([0, \infty])$-measurable?
Clearly, Tonelli's theorem would imply that for every $s \in [0, \infty)$ the mapping $$ [0, \infty) \ni t \mapsto \int_{\mathbb{R}} f(x,t) \mathbb{P}_{\color{red}s}(dx) \tag{1} $$ satisfies the aforementioned measurability. But what can be said about the case above? Does one need additional assumptions on the function $f$? Is it maybe possible to show that the mapping in $(1)$ is continuous?
Some further thoughts:
Assume that the stochastic process $(X_t)_{t \geq 0}$ has (right-)continuous sample paths and the law of $X_t$ is given by $\mathbb{P}_t$, $t \geq 0$. The mapping $$ [0, \infty) \times \Omega \ni (t, \omega) \mapsto X_t (\omega) $$ is jointly measurable. We can further observe that the mapping $$ [0, \infty) \times \Omega \ni (t, \omega) \mapsto ( X_t (\omega), t) $$ is also jointly measurable (since every component is jointly measurable). It then follows that the composition $f(X_t(\omega), t)$ is jointly measurable. We can then apply Tonelli's theorem to conclude that $$ [0, \infty) \ni t \mapsto \mathbb{E}[f(X_t, t)] = \int_{\mathbb{R}} f(x,t) \mathbb{P}_t(dx) $$ is measurable. So the question is whether such a stochastic process always exists. Could the continuity in the Wasserstein metric (which also implies weak convergence) be helpful?