  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. Remarkably, the counterexample comes from classical
theory, where the proposed notion of causal influence determines a redefinition
of the neighbourhood of a cell in cellular automata. 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 no
interaction without disturbance, under which we prove that signalling and
causal influence coincide. The proposed definition has interesting consequences
on the analysis of causal networks, and leads to a revision of the notion of
neighbourhood for classical cellular automata, clarifying a puzzle regarding
their quantisation that apparently makes the neighbourhood larger than the
original one.
