We develop a new nonlinear method to model structure formation in general relativity from a generalization of the relativistic Lagrangian perturbation schemes, controlled by Szekeres (and LTB) exact solutions. The overall approach can be interpreted as the evolution of a deformation field on an inhomogeneous reference model, obeying locally Friedmann-like equations. In the special case of locally one-dimensional deformations, the new model contains the entire Szekeres family of exact solutions. As thus formulated, this approach paraphrases the Newtonian and relativistic Zel'dovich approximations, having a large potential for applications in contexts where relativistic degrees of freedom are relevant. Numerical simulations are implemented to illustrate the capabilities and accuracy of the model.