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.

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