We present our generalisation of a model formalism that allows the derivation of macroscopic diffusion properties of a crystalline material from jump rates of individual atoms or ions. This work is based on a mathematical formalism modelling the uncorrelated motion of particles on a lattice by a Markov chain, from which a master equation in time and space is constructed. This approach is discussed, reformulated, and generalised to be applicable to any three-dimensional crystal system. Specifically, it is capable of describing the diffusion and drift of particles in a tilted potential landscape, as e.g. induced by electric fields. We sketch multiple use cases for systems involving point defects and grain boundaries and use the derived framework to discuss the diffusion of oxygen vacancies in strontium titanate.