Conditions for continuity of the map $x \mapsto \mu_x$ (where $ \mu = \intop \mu_x d \mu (x)$ is the ergodic decomposition of $\mu$)

Question

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system on a Borel space. Let $\{\mu_x\}_x$ be the conditional measures from the ergodic decomposition of $\mu$ (that is, $ \mu = \intop \mu_x d \mu (x)$ where $\mu$-a.e. x $\mu_x$ is invariant and ergodic). Consider the space of probability measures $\mathcal{P}(X)$ with the weak topology. Are there conditions under which the map $x\mapsto\mu_x$ is continuous?

P.S.- I'm also interested in such results w.r.t some other topologies on $\mathcal{P}(X)$ rather than the weak topology (in particular topologies that induced by metrics, but not necessarily Lévy–Prokhorov metric).