  In this paper, we establish that if a sequence of electrical networks
equipped with natural measures converges in the local Gromov-Hausdorff-vague
topology and satisfies certain non-explosion and metric-entropy conditions,
then the sequence of associated discrete-time Markov chains and their local
times also converges. This result applies to many examples, such as critical
Galton-Watson trees, uniform spanning trees, random recursive fractals, the
critical Erd\"os-R\'enyi random graph, the configuration model, and the random
conductance model on fractals. To obtain the convergence result, we
characterize and study extended Dirichlet spaces associated with resistance
forms, and we study traces of electrical networks.
