  We investigate the gravitational backreaction, generated by coupling a
general conformal sector to external, classical gravity, as described by a
conformal anomaly effective action. We address the issues raised by the
regularization of the topological Gauss-Bonnet and Weyl terms in these actions
and the use of dimensional regularization (DR). We discuss both their local and
nonlocal expressions, as possible IR and UV descriptions of conformal theories,
below and above the conformal breaking scale. Our discussion overlaps with
several recent studies of dilaton gravities - obtained via a certain singular
limit of the Einstein-Gauss-Bonnet (EGB) theory - originally introduced as a
way to bypass Lovelock's theorem. We show that nonlocal, purely gravitational
realizations of such EGB theories, quadratic in the dilaton field, beside their
local quartic forms, are possible by a finite renormalization of the Euler
density. Such nonlocal versions, which are deprived of any scale, can be
expanded, at least around flat space, in terms of the combination $R \Box^{-1}$
times multiple variations of the anomaly functional, as pointed out in recent
studies at $d=4$. Similar conclusions can be drawn for the proposed nonlocal
EGB theory. The expansion emerges from previous investigations of the anomalous
conformal Ward identities that constrain such theories around the flat
spacetime limit in momentum space.
