  We develop rigorous notions of causality and causal separability in the
process framework introduced in [Oreshkov, Costa, Brukner, Nat. Commun. 3, 1092
(2012)], which describes correlations between separate local experiments
without a prior assumption of causal order between them. We consider the
general multipartite case and take into account the possibility for dynamical
causal order, where the order of a set of events can depend on other events in
the past. Starting from a general definition of causality, we derive an
iteratively formulated canonical decomposition of multipartite causal
processes, and show that for a fixed number of settings and outcomes for each
party, the respective correlations form a polytope whose facets define causal
inequalities. In the case of quantum processes, we investigate the link between
causality and the theory-dependent notion of causal separability, which we here
extend to the multipartite case based on concrete principles. We show that
causality and causal separability are not equivalent in general by giving an
example of a physically admissible tripartite quantum process that is causal
but not causally separable. We also show that there exist causally separable
(and hence causal) quantum processes that become non-causal if extended by
supplying the parties with entangled ancillas. This example of activation of
non-causality motivates the concepts of extensibly causal and extensibly
causally separable (ECS) processes, for which the respective property remains
invariant under extension with arbitrary ancillas. We characterize the class of
tripartite ECS processes in terms of simple conditions on the form of the
process matrix, which generalize the form of bipartite causally separable
process matrices. We show that the processes realizable by classically
controlled quantum circuits are ECS and conjecture that the reverse also holds.
