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We address the question of numerically simulating the coupling of diffusion, advection and one-speed linear transport, with a specific focus on managing geometrical complexity. We base our work on recent advances from the computer graphics community, which has developed Monte Carlo algorithms simulating linear radiation transport in physically realistic scenes, with numerical costs that remain unaffected by geometrical refinement: adding more details to the scene description does not impact the computation time. The resulting benefits in terms of engineering flexibility are already fully integrated into the cinema industry and are gradually being adopted by the video game industry. Here we demonstrate that the same insensitivity to the geometric complexity can be achieved when considering not only one-speed linear transport, but also its coupling with diffusion and advection. In this case, pure linear-transport paths are replaced with advection-diffusion/linear-transport paths, which are composed of subpaths. Each subpath represents one of the three physical phenomena, and coupling is handled by switching from one subpath (i.e. phenomenon) to another. This approach is illustrated using a porous medium involving up to 10,000 pores, with the computation time being strictly independent of the number of pores, showing its ability to facilitate engineering calculations in complex geometries.