We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension conditions.
"Convergence of Brownian motions on metric measure spaces under Riemannian Curvature–Dimension conditions." Electron. J. Probab. 24 1 - 36, 2019. https://doi.org/10.1214/19-EJP346