The ability to use external magnetic fields to influence the microstructure in polycrystalline materials has potential applications in microstructural engineering. To explore this potential and to understand the complex interactions between electromagnetic fields and solid-state matter transport we consider a phase-field-crystal (PFC) model that captures the basic physics of magnetocrystalline interactions. After investigations of a PFC model to study grain growth without an external magnetic field, the thesis concentrates on the influence of magnetic fields in the PFC model, to understand the basic phenomena. The second part considers a coarse graining of the PFC model towards an amplitude expansion (APFC) model to enable 3D simulations. The coupling with magnetic fields allows together with efficient and scalable numerical algorithms to examine the role of external magnetic fields on the evolution of defect structures and grain boundaries, on diffusion time scales. Large scale simulations in 2D and 3D also allow to obtain statistical data on grain growth under the influence of external fields and to validate with experimental data.