  In this paper, we formulate a generalised photon Boltzmann hierarchy that
allows us to model the evolution and creation of spectral distortion
anisotropies in the early Universe. We directly build on our first paper in
this series, extending the thermalisation Green's function treatment to the
anisotropic case. We show that the problem can be described with the common
Boltzmann hierarchy for the photon field extended by new spectral parameters --
a step that reduces the complexity of the calculation by at least two orders of
magnitude. Our formalism describes the effects of i) Doppler and potential
driving, ii) spectral evolution by Compton scattering, iii) perturbed
thermalisation and iv) anisotropic heating on the distortion anisotropies. We
highlight some of the main physical properties of the equations and also
outline the steps for computing CMB power spectra including distortion
anisotropies. Limitations and extensions of the formulation are also briefly
discussed. The novel Boltzmann hierarchy given here is the basis for a series
of companion papers studying how distortion anisotropies evolve in the
perturbed Universe and which physical processes could be constrained using
future CMB imaging techniques.

Cosmology and Nongalactic Astrophysics