Electromigration (EM) is the motion of lattice atoms under high-density electric current. It is an important mechanism for structural changes in nanoelectronic devices and a major contributor to the electroplastic effect in structural metals. Recently, a phase-field-crystal (PFC) model for studying EM in metals was developed and shown to successfully capture many important EM-driven structural evolutions at experimentally relevant timescales. In this work, connections between the previous PFC EM model and existing EM theories are established. It is shown that the PFC model can be linked to both the electron-density-based quantum EM theory and the drift-diffusion-based continuum EM theory, therefore filling an important gap in the theoretical and computational study of EM-related phenomena. The numerical method for implementing the PFC EM model is discussed in detail. The well-established Blech effect is quantitatively reproduced using the model.