The migration process of magnetic nanoparticles and colloids in solution under the influence of magnetic field gradients, which is also known as magnetophoresis, is an essential step in the separation technology used in various biomedical and engineering applications. Many works have demonstrated that in specific situations, separation can be performed easily with the weak magnetic field gradients created by permanent magnets, a process known as low-gradient magnetic separation (LGMS). Due to the level of complexity involved, it is not possible to understand the observed kinetics of LGMS within the classical view of magnetophoresis. Our experimental and theoretical investigations in the last years unravelled the existence of two novel physical effects that speed up the magnetophoresis kinetics and explain the observed feasibility of LGMS. Those two effects are (i) cooperative magnetophoresis (due to the cooperative motion of strongly interacting particles) and (ii) magnetophoresis-induced convection (fluid dynamics instability originating from inhomogeneous magnetic gradients). In this feature article, we present a unified view of magnetophoresis based on the extensive research done on these effects. We present the physical basis of each effect and also propose a classification of magnetophoresis into four distinct regimes. This classification is based on the range of values of two dimensionless quantities, namely, aggregation parameter N* and magnetic Grashof number Gr(m), which include all of the dependency of LGMS on various physical parameters (such as particle properties, thermodynamic parameters, fluid properties, and magnetic field properties). This analysis provides a holistic view of the classification of transport mechanisms in LGMS, which could be particularly useful in the design of magnetic separators for engineering applications.