  Given a topological dynamical system $(X,T)$, we study properties of the mean
orbital pseudo-metric $\bar E$ defined by \[ \bar E(x,y)= \limsup_{n\to\infty }
\min_{\sigma\in S_n}\frac{1}{n}\sum_{k=0}^{n-1}d(T^k(x),T^{\sigma(k)}(y)), \]
where $x,y\in X$ and $S_n$ is the permutation group of $\{0,1,\ldots,n-1\}$.
Let $\hat\omega_T(x)$ denote the set of measures quasi-generated by a point
$x\in X$. We show that the map $x\mapsto\hat\omega_T(x)$ is uniformly
continuous if $X$ is endowed with the pseudo-metric $\bar E$ and the space of
compact subsets of the set of invariant measures is considered with the
Hausdorff distance. We also obtain a new characterisation of $\bar
E$-continuity, which connects it to other properties studied in the literature,
like continuous pointwise ergodicity introduced by Downarowicz and Weiss.
Finally, we apply our results to reprove some known results on $\bar
E$-continuous and mean equicontinuous systems.
