The variation of the physical conditions across the three dimensions of our Galaxy is a major source of complexity for the modelling of the foreground signal facing the cosmic microwave background (CMB). In the present work, we demonstrate that the spin-moment expansion formalism provides a powerful framework to model and understand this complexity, and we put special focus on the effects that arise from variations of the physical conditions along each line of sight on the sky. We performed the first application of the moment expansion to reproduce a thermal dust model largely used by the CMB community, demonstrating its power as a minimal tool to compress, understand, and model the information contained within any foreground model. Furthermore, we used this framework to produce new models of thermal dust emission containing the maximal amount of complexity allowed by the current data while remaining compatible with the observed angular power spectra by the Planck mission. By assessing the impact of these models on the performance of component separation methodologies, we conclude that the additional complexity contained within the third dimension could represent a significant challenge for future CMB experiments and that different component separation approaches are sensitive to different properties of the moments.
