  The ghost-gluon scattering kernel is a special correlation function that is
intimately connected with two fundamental vertices of the gauge sector of QCD:
the ghost-gluon vertex, which may be obtained from it through suitable
contraction, and the three-gluon vertex, whose Slavnov-Taylor identity contains
that kernel as one of its main ingredients. In this work we present a detailed
nonperturbative study of the five form factors comprising it, using as starting
point the `one-loop dressed' approximation of the dynamical equations governing
their evolution. The analysis is carried out for arbitrary Euclidean momenta,
and makes extensive use of the gluon propagator and the ghost dressing
function, whose infrared behavior has been firmly established from a multitude
of continuum studies and large-volume lattice simulations. In addition, special
Ans\"atze are employed for the vertices entering in the relevant equations, and
their impact on the results is scrutinized in detail. Quite interestingly, the
veracity of the approximations employed may be quantitatively tested by
appealing to an exact relation, which fixes the value of a special combination
of the form factors under construction. The results obtained furnish the two
form factors of the ghost-gluon vertex for arbitrary momenta, and, more
importantly, pave the way towards the nonperturbative generalization of the
Ball-Chiu construction for the longitudinal part of the three-gluon vertex.
