  We prove a quantitative, finitary version of Trofimov's result that a
connected, locally finite vertex-transitive graph G of polynomial growth admits
a quotient with finite fibres on which the action of Aut(G) is virtually
nilpotent with finite vertex stabilisers. We also present some applications. We
show that a finite, connected vertex-transitive graph G of large diameter
admits a quotient with fibres of small diameter on which the action of Aut(G)
is virtually abelian with vertex stabilisers of bounded size. We also show that
G has moderate growth in the sense of Diaconis and Saloff-Coste, which is known
to imply that the mixing and relaxation times of the lazy random walk on G are
quadratic in the diameter. These results extend results of Breuillard and the
second author for finite Cayley graphs of large diameter. Finally, given a
connected, locally finite vertex-transitive graph G exhibiting polynomial
growth at a single, sufficiently large scale, we describe its growth at
subsequent scales, extending a result of Tao and an earlier result of our own
for Cayley graphs. In forthcoming work we will give further applications.
