Causal mediation analysis is advantageous for mechanism investigation. In settings with multiple causally ordered mediators, path-specific effects have been introduced to specify the effects of certain combinations of mediators. However, most path-specific effects are unidentifiable. An interventional analog of path-specific effects is adapted to address the non-identifiability problem. Moreover, previous studies only focused on cases with two or three mediators due to the complexity of the mediation formula in a large number of mediators. In this study, we provide a generalized definition of traditional path-specific effects and interventional path-specific effects with a recursive formula, along with the required assumptions for nonparametric identification. Subsequently, a general approach is developed with an arbitrary number of multiple ordered mediators and with time-varying confounders. All methods and software proposed in this study contribute to comprehensively decomposing a causal effect confirmed by data science and help disentangling causal mechanisms in the presence of complicated causal structures among multiple mediators.