This question is an extension of Feller continuity of the stochastic kernel. Nate Eldredge provided a nice counterexample, but I failed trying to extend it to the compact set $B$.
The setting is the same: we are given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for each $x$, i.e. $K(x,X) = 1$.
Let $f:X\to \mathbb R$. We say that $f\in \mathcal C(B)$ if $f$ is continuous and bounded on $B$.
Weak Feller continuity of $K$ means that if $f\in\mathcal C(X)$ then $F\in\mathcal C(X)$ where $$ F(x):=\int\limits_X f(y)K(x,dy). $$
I wonder if it implies that if $g\in \mathcal C(B)$ then $$ G(x):=\int\limits_Bg(y)K(x,dy) $$ also belongs to $\mathcal C(B)$ for any compact set $B$?