The fourth review discusses hydrodynamics problems of gas fluidized moving magnetizable solids. Various versions of moving beds are considered highlighting their hydrodynamic problems. The main targets of the discussion are the bed mechanics, magnetic field action (magnetization modes) and the bed rheology. The paper intends to analyze various devices and ideas investigated over 20 years since the 1980s. The approach of data treatment followed in the previous reviews yields a unified analysis which establishes relationships between the results of different variants of moving fluidized beds controlled by external magnetic fields.