  It was recently suggested that causal structures are both dynamical, because
of general relativity, and indefinite, due to quantum theory. The process
matrix formalism furnishes a framework for quantum mechanics on indefinite
causal structures, where the order between operations of local laboratories is
not definite (e.g. one cannot say whether operation in laboratory A occurs
before or after operation in laboratory B). Here we develop a framework for
"dynamics of causal structures", i.e. for transformations of process matrices
into process matrices. We show that, under continuous and reversible
transformations, the causal order between operations is always preserved.
However, the causal order between a subset of operations can be changed under
continuous yet nonreversible transformations. An explicit example is that of
the quantum switch, where a party in the past affects the causal order of
operations of future parties, leading to a transition from a channel from A to
B, via superposition of causal orders, to a channel from B to A. We generalise
our framework to construct a hierarchy of quantum maps based on transformations
of process matrices and transformations thereof.
