  We develop the transport operator formalism, a new line-of-sight integration
framework to calculate the anisotropies of the Cosmic Microwave Background
(CMB) at the linear and non-linear level. This formalism utilises a
transformation operator that removes all inhomogeneous propagation effects
acting on the photon distribution function, thus achieving a split between
perturbative collisional effects at recombination and non-perturbative
line-of-sight effects at later times. The former can be computed in the
framework of standard cosmological perturbation theory with a second-order
Boltzmann code such as SONG, while the latter can be treated within a separate
perturbative scheme allowing the use of non-linear Newtonian potentials. We
thus provide a consistent framework to compute all physical effects contained
in the Boltzmann equation and to combine the standard remapping approach with
Boltzmann codes at any order in perturbation theory, without assuming that all
sources are localised at recombination.
