We establish that if a sequence of electrical networks equipped with conductance 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 conditioned on size, uniform spanning trees, random recursive fractals, the critical Erdős--Rényi 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.