Suppose we have a locally compact separable space $X$, on which we define a Markov kernel $P:X\times\mathcal B(X)\to[0,1]$, which satisfies the weak Feller property, that is to say, for any $f\in C_b(X)$, the function $$ Pf(y)=\int_X\! P(y,\mathrm dx)f(x) $$ is continuous in $y$. Now suppose there exists an invariant measure $\pi\in\mathcal P(X)$, i.e. $P^n(x,\cdot)\to\pi$ weakly for all $x$. Then the book by Meyn and Tweedie claims in Prop 6.4.2 that $$P^nf\to\pi(f)$$ uniformly on compact sets for any $f\in C_b(X)$. However, I don't see how to get anything better than pointwise convergence and uniform boundedness here.