Let $E$ be a locally compact Hausdorff space with countable base and $X_t$ be a stochastic process taking values in the one-point compactification of $E$ (with the Borel $\sigma$-algebra). Let $f$ be a continuous function vanishing at infinity. I'm wondering under what conditions it is true that $x \mapsto \mathbf{E}^{x}[f(X_t)]$ is a continuous function?

If $X_t$ is a Brownian motion on $\mathbb{R}$, it is straightforward to verify this is true and, in fact, it is true whenever $X_t$ is an Ito diffusion.

An example where this fails is if we let $E=\mathbf{R}$ and let $X_t$ be a reflected Brownian motion on $\{ x: x\geq 0 \}$ and be the negative of the absolute value of a Brownian motion in $\mathbb{R}^3$ on $\{x : x<0 \}$.

More broadly, I'm curious as to what kind of conditions on the sample paths of a Markov process $X$ with continuous paths force it to be a Feller process. The condition I'm asking about seems to be the one that does not come for free if you start with such an $X$.