It has long been one of the main challenges in science and engineering to capture the probabilistic response of high-dimensional nonlinear stochastic dynamic systems involving double randomness, i.e. randomness in both system parameters and excitations. For this purpose, a globally-evolving-based generalized density evolution equation (GE-GDEE) is established. Generally, for a multi-dimensional nonlinear system involving double randomness, if one single physical quantity as a response of the system is of interest, a GE-GDEE, as a two-dimensional partial differential equation (PDE) governing the probability density function (PDF), can be derived. The effective drift coefficients, which represent the physically driving force function in the GE-GDEE, can be determined based on the data from some representative deterministic dynamic analyses of the underlying physical system. A new estimator for effective drift coefficients is developed based on the vine copulas. Once the effective drift coefficients are determined, the GE-GDEE can be solved to capture the probability distributions of the quantities of interest. Several numerical examples, including linear and nonlinear multi-degree-of-freedom (MDOF) systems subjected to white noise or non-stationary earthquake ground motions, are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.