  We prove that, if $G$ is a second-countable topological group with a
compatible right-invariant metric $d$ and $(\mu_{n})_{n \in \mathbb{N}}$ is a
sequence of compactly supported Borel probability measures on $G$ converging to
invariance with respect to the mass transportation distance over $d$ and such
that $\left(\mathrm{spt} \, \mu_{n}, d\!\!\upharpoonright_{\mathrm{spt} \,
\mu_{n}}, \mu_{n}\!\!\upharpoonright_{\mathrm{spt} \, \mu_{n}}\right)_{n \in
\mathbb{N}}$ concentrates to a fully supported, compact $mm$-space
$\left(X,d_{X},\mu_{X}\right)$, then $X$ is homeomorphic to a $G$-invariant
subspace of the Samuel compactification of $G$. In particular, this confirms a
conjecture by Pestov and generalizes a well-known result by Gromov and Milman
on the extreme amenability of topological groups. Furthermore, we exhibit a
connection between the average orbit diameter of a metrizable flow of an
arbitrary amenable topological group and the limit of Gromov's observable
diameters along any net of Borel probability measures UEB-converging to
invariance over the group.
