To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of finite-dimensional distributions. Thus in many settings, weak convergence to a continuous limit process can be deduced almost immediately from convergence of finite-dimensional distributions. These results may be technically useful in simplifying proofs of weak convergence, particularly in infinite-dimensional settings. The results rely on a technical tightness condition involving stopping times and predictability of imminent jumps.
David Aldous. "Stopping Times and Tightness. II." Ann. Probab. 17 (2) 586 - 595, April, 1989. https://doi.org/10.1214/aop/1176991417