A characteristic dependence of voltage control of perpendicular magnetic anisotropy (VCMA) on oxygen migration at Fe/MgO interfaces was revealed by performing systematic $ab\ initio$ study of the energetics of the oxygen path around the interface. We find that the surface anisotropy energy exhibits a Boltzmann sigmoidal behavior as a function of the migrated O-atoms concentration. The obtained variation of the VCMA efficiency factor $\beta$ reveals a saturation limit beyond a critical concentration of migrated O, about 54%, at which the anisotropy switches from perpendicular to in plane. Furthermore, depending on the range of variation of the applied voltage, two regimes associated with reversible or irreversible ions displacement are predicted to occur, yielding different VCMA response. According to our findings, one can distinguish from the order of magnitude of $\beta$ the VCMA driving mechanism: an effect of several tens of fJ/(V.m) is likely associated to charge-mediated effect combined with slight reversible oxygen displacements whereas an effect of the order of thousands of fJ/(V.m) is more likely associated with irreversible oxygen ionic migration.