  Interacting many-body systems with explicitly accessible spatio-temporal
correlation functions are extremely rare, especially in the absence of
integrability. Recently, we identified a remarkable class of such systems and
termed them dual-unitary quantum circuits. These are brick-wall type local
quantum circuits whose dynamics are unitary in both time and space. For these
systems the spatio-temporal correlation functions are non-trivial only at the
edge of the causal light cone and can be computed in terms of one-dimensional
transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree
of generality of the observed dynamical features remained unclear. Here we
address this question by introducing arbitrary perturbations of the local
gates. Considering fixed perturbations, we prove that for a particular class of
unperturbed elementary dual-unitary gates the correlation functions are still
expressed in terms of one-dimensional transfer matrices. These matrices,
however, are now contracted over generic paths connecting the origin to a fixed
endpoint inside the causal light cone. The correlation function is given as a
sum over all such paths. Our statement is rigorous in the "dilute limit", where
only a small fraction of the gates is perturbed, and in the presence of random
longitudinal fields, but we provide theoretical arguments and stringent
numerical checks supporting its validity even in the clean case and when all
gates are perturbed. As a byproduct, in the case of random longitudinal fields
-- which turns out to be equivalent to certain classical Markov chains -- we
find four types of non-dual-unitary(and non-integrable) interacting many-body
systems where the correlation functions are exactly given by the path-sum
formula.
