  Given a sequence of resistance forms that converges with respect to the
Gromov-Hausdorff-vague topology and satisfies a uniform volume doubling
condition, we show the convergence of corresponding Brownian motions and local
times. As a corollary of this, we obtain the convergence of time-changed
processes. Examples of our main results include scaling limits of Liouville
Brownian motion, the Bouchaud trap model and the random conductance model on
trees and self-similar fractals. For the latter two models, we show that under
some assumptions the limiting process is a FIN diffusion on the relevant space.
