Any triangular array of row independent random vectors with continuous df's has a standard reduction to random vectors with values in the unit cube. The reduced empirical process belonging to the transformed random vectors is always relatively compact. Weak convergence to a (necessarily Gaussian) process holds iff the corresponding covariance kernel converges pointwise.
"Convergence of the Reduced Empirical Process for Non-I.I.D. Random Vectors." Ann. Statist. 3 (2) 528 - 531, March, 1975. https://doi.org/10.1214/aos/1176343084