We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and (µ_n)nεℕ 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 (sptµn, d{up harpoon right}sptµ_n,µ_n{up harpoon right}sptµ_n)nεN concentrates to a fully supported, compact mm-space (X, dX,µX), 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.