  We derive the conformal constraints satisfied by classical vertices of a
(Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at
$d=4$ (4d EGB). In $4d$ EGB they are obtained by a singular limit of the
integral of the Euler-Poincar\`e density. Our analysis exploits the relation
between this theory and the conformal anomaly action, which allows to uncover
some interesting features of the GB vertex at cubic and quartic level. If we
introduce a conformal decomposition of the metric, the resulting theory can be
formulated in two different versions, which are regularization dependent, a
local one which is quartic in the dilaton field, and a nonlocal one, with a
quadratic dilaton. The nonlocal version is derived by a finite redefinition of
the GB density with the inclusion of a $(d-4)R^2$ correction, before performing
the singular $d\to 4$ limit. In the local version of the theory, we show how
the independent dynamics of the metric and of the dilaton are interwinded by a
classical trace identity. Three-gravitational wave interactions can be
organised in a nontrivial way by using directly the nonlocal 4d EGB version of
the theory. This is possible thanks to the consistency of such formulation -
only up to 3-point functions - directly inherited from the conformal anomaly
(Riegert) action. The constraints satisfied by the vertices are classical,
hierarchical Ward identities. At quartic level, similar relations are derived,
borrowing from the analysis of the counterterms of the 4T correlators of the
conformal anomaly action, as defined by a perturbative expansion. For $d\neq 4$
these constraints hold also for Lovelock actions. They can be extended to
higher order topological invariants in such class of theories.
