The Newtonian Lagrangian perturbation theory is a widely used framework to
study structure formation in cosmology in the nonlinear regime. We review a
general-relativistic formulation of such a perturbation approach, emphasizing
results on already developed extensive formalism including among other aspects:
the non-perturbative modeling of Ricci and Weyl curvatures, gravitational waves
and pressure-supported fluids. We discuss subcases of exact solutions related
to Szekeres Class II and, as exact average model, Ricci-flat LTB models. This
latter forms the basis of a generalization that we then propose in terms of a
scheme that goes beyond the relativistic Lagrangian perturbation theory on a
global homogeneous-isotropic background cosmology. This new approximation does
not involve a homogeneous reference background and it contains Szekeres class I
(and thus general LTB models) as exact subcases. Most importantly, this new
approximation allows for the interaction of structure with an evolving
`background cosmology’, conceived as a spatial average model, and thus includes
cosmological backreaction.
