 The concept of causal non-separability has been introduced to describe physical processes that follow quantum theories' local laws but cannot be embedded in a well-defined global causal structure. While there are two proposed generalizations for multipartite scenarios, they are inequivalent. A new definition is proposed for the general case, which turns out to be equivalent to extensible causal non-separability and is more natural for multipartite scenarios. Necessary and sufficient conditions are derived to characterize causally, non-separable processes in practice, allowing practical tests through witnesses of causal non-separability. This article was offered by Julian Wex, Alistair A. Abbott and Cyril Bransard.