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 \(d\) 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.