We consider boundary-layer structures that arise in connection with the transport of neutral particles (e.g., photons or neutrons) through a participating medium. Such boundary-layer structures were previously identified by the authors in certain particular cases [Phys. Rev. E 104, L032801 (2021)2470-004510.1103/PhysRevE.104.L032801]. Extending the previous work to anisotropic scattering and general Fresnel boundary conditions, this contribution presents computational algorithms which (1) resolve the aforementioned layers as well as previously unreported boundary layers associated with Fresnel boundary transmission and reflection, and (2) yield accurate simulations at fixed computational cost for transport under phase functions with arbitrarily strong anisotropy. The present paper additionally includes (3) Mathematical proofs which justify the numerical methods proposed for resolution of boundary-layer structures. The impact of the new theory on algorithmic performance is demonstrated through a series of 1D computational benchmarks that emulate typical photon- and neutron-transport applications such as, e.g., optical tomography, and nuclear reactor analysis and design. Experimental results for transmission of photons through turbid media are presented, exhibiting close agreement between simulated and experimental data. As illustrated by means of a variety of numerical results, the proposed boundary-layer-based approach tackles transport problems with unprecedented accuracy and efficiency.