  We derive and analyze the conformal Ward identities (CWI's) of a tensor
4-point function of a generic CFT in momentum space. The correlator involves
the stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We extend
the reconstruction method for tensor correlators from 3- to 4-point functions,
starting from the transverse traceless sector of the $TOOO$. We derive the
structure of the corresponding CWI's in two different sets of variables,
relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2
(graviton + scalar $\to$ two scalars) scattering processes. The equations are
all expressed in terms of a single form factor. In both cases, we discuss the
structure of the equations and their possible behaviors in various asymptotic
limits of the external invariants. A comparative analysis of the systems of
equations for the $TOOO$ and those for the $OOOO$, both in the general
(conformal) and dual-conformal/conformal (dcc) cases, is presented. We show
that in all the cases the Lauricella functions are homogenous solutions of such
systems of equations, also described as parametric 4K integrals of modified
Bessel functions.
