Uniform rank gradient, cost and local-global convergence

The notion of combinatorial cost for sequences of graphs was introduced by Elek as an analogue of the cost of measure preserving equivalence relations. We show that if a graph sequence is local-global convergent, then its combinatorial cost equals the cost of the limit graphing.
This in particular implies previous results of Elek on combinatorial cost, and gives an alternate proof of the Abert-Nikolov theroem that connects the rank gradient of a chain of subgroups to the cost of its profinite completion.
It also turns out that local-global convergence is a useful tool in reinforcing previous results on the rank gradient. We obtain a uniform continuity result for the rank gradient for Farber sequences of subgroups in groups with fixed price, and show vanishing of the rank gradient in finitely presented amenable groups for arbitrary sequences (with index tending to infinity).