A Cayley diagram for a Cayley graph G=Cay(Γ,E)G=Cay(Γ,E) is an edge labelling of GG with generators from EE so that a path is labelled with a relation in ΓΓ if and only if it is a cycle. I will show how Aut(G)Aut(G)-f.i.i.d. Cayley diagrams can be used to lift ΓΓ-f.i.i.d. solutions of local combinatorial problems to Aut(G)Aut(G)-f.i.i.d. solutions. And, I will investigate which graphs admit Aut(G)Aut(G)-f.i.i.d. Cayley diagrams, answering a question of Weilacher on approximate chromatic numbers in the process.