  This work presents a set of coupled Boltzmann equations describing the
intensity and polarisation Stokes parameters of the SGWB. Collision terms,
which account for gravitational Compton scattering off of massive objects, are
also included. This set resembles that for the CMB Stokes parameters, but the
different spin nature of the gravitational radiation and the physics involved
in the scattering process determine crucial differences. In this case, due to
the Rutherford angular dependence of the cross section, all the SGWB intensity
multipoles of order $\ell$ are scattered out, producing outgoing intensity
anisotropies of any order $\ell$ if they are present in the incoming radiation.
On the other hand, SGWB linear polarisation modes can be expanded in a basis of
spherical harmonics with $m=\pm 4$ and $\ell\ge 4$. This means that SGWB
polarisation modes can be generated from unpolarised anisotropic radiation only
with $m=\pm 4$, therefore requiring at least a hexadecapole anisotropy
($\ell\ge 4$) in the incoming intensity. Assuming a simplified toy model, we
solve analytically the set of coupled Boltzmann equations to get explicit
expressions for the intensity and polarisation angular power spectra. We
confirm the contribution of the gravitational Compton scattering to the SGWB
anisoptropies is extremely small for collisions with compact objects in the
frequency range of current and upcoming surveys. The system of coupled
Boltzmann equations presented here provides a way to an accurate estimate of
the total amount of anisotropies generated by multiple SGWB scattering
processes off of massive objects, as well as the interplay between polarisation
and intensity, during the GW propagation across the LSS of the universe. These
results will be useful for the full treatment of the astrophysical SWGB
anisotropies in view of upcoming gravitational waves observatories.
