The advent of counterfactual-based mediation analysis has triggered enormous progress on how, and under what assumptions, one may disentangle path-specific effects upon combining arbitrary (possibly nonlinear) models for mediator and outcome. However, current developments have largely focused on single mediators because required identification assumptions prohibit simple extensions to settings with multiple mediators that may depend on one another. In this article, we propose a procedure for obtaining fine-grained decompositions that may still be recovered from observed data in such complex settings. We first show that existing analytical approaches target specific instances of a more general set of decompositions and may therefore fail to provide a comprehensive assessment of the processes that underpin cause-effect relationships between exposure and outcome. We then outline conditions for obtaining the remaining set of decompositions. Because the number of targeted decompositions increases rapidly with the number of mediators, we introduce natural effects models along with estimation methods that allow for flexible and parsimonious modeling. Our procedure can easily be implemented using off-the-shelf software and is illustrated using a reanalysis of the World Health Organization's Large Analysis and Review of European Housing and Health Status (WHO-LARES) study on the effect of mold exposure on mental health (2002-2003).