  We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations
to first order in quantities small compared to the Fermi energy, valid for
Dirac edge and surface electrons with spin-momentum locking, as relevant for
topological insulators. We discuss in detail several of the key technical
points and assumptions of the derivation, and provide a Riccati-parametrization
of the equations. Solving first the equilibrium equations for S/N and S/F
bilayers and Josephson junctions, we study the superconducting proximity effect
in Dirac materials. Similarly to related works, we find that the effect of an
exchange field depends strongly on the direction of the field. Only components
normal to the transport direction lead to attenuation of the Cooper pair
wavefunction inside the F. Fields parallel to the transport direction lead to
phase-shifts in the dependence on the superconducting phase difference for both
the charge current and density of states in an S/F/S-junction. Moreover, we
compute the differential conductance in S/N and S/F bilayers with an applied
voltage bias, and determine the dependence on the length of the N and F regions
and the exchange field.
