  We elaborate on the structure of the conformal anomaly effective action up to
4-th order, in an expansion in the gravitational fluctuations $(h)$ of the
background metric, in the flat spacetime limit. For this purpose we discuss the
renormalization of 4-point functions containing insertions of stress-energy
tensors (4T), in conformal field theories in four spacetime dimensions with the
goal of identifying the structure of the anomaly action. We focus on a
separation of the correlator into its transverse/traceless and longitudinal
components, applied to the trace and conservation Ward identities (WI) in
momentum space. These are sufficient to identify, from their hierarchical
structure, the anomaly contribution, without the need to proceed with a
complete determination of all of its independent form factors. Renormalization
induces sequential bilinear graviton-scalar mixings on single, double and
multiple trace terms, corresponding to $R\square^{-1}$ interactions of the
scalar curvature, with intermediate virtual massless exchanges. These
dilaton-like terms couple to the conformal anomaly, as for the chiral anomalous
WIs. We show that at 4T level a new traceless component appears after
renormalization. We comment on future extensions of this result to more general
backgrounds, with possible applications to non local cosmologies.
