  We study the relation of causal influence between input systems of a
reversible evolution and its output systems, in the context of operational
probabilistic theories. We analyse two different definitions that are borrowed
from the literature on quantum theory -- where they are equivalent. One is the
notion based on signalling, and the other one is the notion used to define the
neighbourhood of a cell in a quantum cellular automaton. The latter definition,
that we adopt in the general scenario, turns out to be strictly weaker than the
former: it is possible for a system to have causal influence on another one
without signalling to it. We stress that, according to our definition, it is
impossible anyway to have causal influence in the absence of an interaction,
e.g.~in a Bell-like scenario. We study various conditions for causal influence,
and introduce the feature that we call {\em no interaction without
disturbance}, under which we prove that signalling and causal influence
coincide.
