The propagating source method has been extended to solve the Boltzmann equation with a quasi-linear diffusion scattering operator. A half-range polynomial expansion method is used to reduce the integral-diffusion form of the ‘collisional’ Boltzmann equation to an infinite set of linear hyperbolic partial differential equations in the harmonics of the polynomial expansion. The lowest-order truncation of the coupled set of equations yields an inhomogeneous form of the well-known telegrapher equation, which, unlike the homogeneous telegrapher equation, does not introduce physically unrealistic pulse solutions. Anisotropic quasi-linear scattering models for which the index $q$ of the power spectrum of magnetic fluctuations satisfies $1\,{<}\,q\,{<}\,2$ admit slow scattering through $90^{\circ}$ and no scattering through $90^{\circ}$ for $q \,{\ge}\,2$. Accordingly, four models that either allow or enhance scattering through $90^{\circ}$ are used to augment the standard quasi-linear model for pitch-angle scattering. These are mirroring, dynamical turbulence and two distinct wave-based models. In the case that mirroring is responsible for scattering particles through $90^{\circ}$, together with the standard QLT (quasi-linear theory) pitch-angle diffusion model for scattering within the forward and backward hemispheres, it is found that the QLT isotropic and anisotropic models are well approximated by relaxation time scattering models. As an application of the general study, the implications of the four models introduced to redress the difficulties faced by QLT in describing scattering through $90^{\circ}$ are briefly considered. An initial beam was found to relax more rapidly for either the dynamical turbulence or wave models with resonant scattering through $90^{\circ}$ than for mirroring models.