Non-linear cosmic structure formation can be described by a kinetic field theory for classical particle ensembles out of equilibrium. Building upon the path-integral formalism for classical mechanics, this theory structurally resembles a statistical quantum field theory. Its generating functional encapsulates the statistical properties of an initial state of the ensemble and the equation of motion of its particles. Interactions between the particles are described by an interaction operator, which can either be Taylor expanded into a perturbation series of Feynman diagrams, or approximated in a mean-field approach. For cosmology, the mean-field approach has proven to be quite successful, allowing to calculate non-linear power spectra for cosmic structures analytically quite deeply into the non-linear regime even at late times and small scales.
The conceptual simplicity and the flexibility of this kinetic field theory allow to apply it quite easily to different models for the dark matter, and to generalisations of gravity theory. Extended model or theory spaces can now be studied with little effort for their effects on non-linear cosmic structure formation. In the talk, I intend to (i) review kinetic field theory, (2) summarise some results for standard cosmology, and (iii) describe extensions towards axionic or axion-like dark-matter models and some generalisations of general relativity.