Dynamic actions such as strong winds and earthquakes often have significant randomness and non-stationarity，which can have disastrous effects on practical engineering structures. Therefore，accurately evaluating the dynamic reliability of high-dimensional nonlinear systems under non-stationary stochastic excitations is crucial for the disaster-resistant design and optimization of these structures. This paper presents a numerical method for solving the high-dimensional nonlinear dynamic reliability under non-stationary noises，based on the globally-evolving-based generalized density evolution equation （GE-GDEE） for generic continuous processes. Specifically，if we are concerned with the first-passage reliability of a quantity of interest within a specified safe domain，an absorbing boundary process （ABP） of the quantity of interest can be constructed. This leads to a two-dimensional partial differential equation for its transient probability density function （PDF），known as the GE-GDEE for ABPs. The effective drift coefficient in the GE-GDEE，which serves as the global physical driving force for evolution of the PDF，can be identified using data from representative deterministic dynamic analyses. The solution for dynamic reliability can be obtained by solving the GE-GDEE. This paper includes two numerical examples to verify the efficiency and accuracy of the proposed method and discusses areas that require further study.