Density functional theory (DFT) methods have been applied to study C₃₂ fullerenes built from four-, five-, and six-membered rings. The relative energies of pure C₃₂ fullerenes have been evaluated to locate three most stable structures, 32:D_4d with two squares, 1:D₃ without square and 5:C_s with one square. Structural analysis reveals that there is a rearrangement pathway between the lowest energy classical isomer 1:D₃ and the lowest energy non-classical isomer 32:D_4d, and 5:C_s behaves just as an intermediate between them. The kinetic processes of generalized Stone-Wales transformation (GSWT) with four-membered rings have been explored and two distinct reaction mechanisms are determined by all the transition states and intrinsic reaction coordinates with PBE1PBE/6-31G(d) approach for the first time. One mechanism is the concerted reaction with a rotating dimer closed to the cage surface and another is the stepwise reaction with a carbene-like sp³ structure, whereas the latter is sorted into two paths based on four-membered ring vanishing before or after the formation of the carbene-like structure. It is indicated that there is no absolute preference for any mechanism, which depends on the adaptability of different reactants on the diverse mechanisms. Furthermore, it's found that the interconversion process with the participation of squares is more reactive than the rearrangement between C₆₀_I_h and C₆₀_C_2v, implying some potential importance of non-classical small fullerenes in the fullerene isomerization.