  We construct a sequence of finite graphs that weakly converge to a Cayley
graph, but there is no labelling of the edges that would converge to the
corresponding Cayley diagram. A similar construction is used to give graph
sequences that converge to the same limit, and such that a spanning tree in one
of them has a limit that is not approximable by any subgraph of the other. We
give an example where this subtree is a Hamiltonian cycle, but convergence is
meant in a stronger sense. These latter are related to whether having a
Hamiltonian cycle is a testable graph property.
