These are the four Mathematica notebooks showing the analytic connection between the inter-band and intra-band kernels for both the Kubo and the Boltzmann expressions in my thesis.
To this end, one constructs a vectorial difference of the two relevant poles in the complex (half) plane, and takes the (angle-resolved) limit.
The inter-band Kubo equations collapse neatly onto the intra-band equations (see Kubo notebooks). The inter-band Boltzmann equations need additional (non Gamma leading) terms for the intra-band equations to emerge. Further, here the intra-band heat-heat kernels require additional terms.