Let G be a finitely generated, virtually nilpotent group. We consider three closely related problems:
(i) description of the asymptotic cone for an equivariant ergodic family of inner metrics on G, generalizing Pansu's theorem;
(ii) existence of the limit shapes for First Passage Percolation for general (not necessarily independent) ergodic process on edges of a Cayley graph of G;
(iii) formulation of a sub-additive ergodic theorem over a general ergodic G-action.
The limiting objects are given in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal'cev completion of G.