During orbital precession minima, the Sahara was humid and hosted tropical plant species thus providing a corridor for Hominins migration. Uncertainties remain over the climatic processes controlling the initiation, demise and amplitude of these African Humid Periods (AHPs). Here we present transient simulations of the penultimate deglaciation and Last Interglacial period (LIG), and compare them to transient simulations of the last deglaciation and Holocene. We find that the strengthening of the Atlantic Meridional Overturning Circulation (AMOC) at the end of the deglacial millennial-scale events exerts a dominant control on the abrupt initiation of AHPs, as the AMOC modulates the position of the Intertropical Convergence Zone (ITCZ). In addition, residual Northern Hemispheric (NH) ice-sheets can delay the AHP peak. Through its impact on NH ice-sheets disintegration and thus AMOC variations, the larger rate of insolation increase during the penultimate compared to the last deglaciation can explain the earlier and more abrupt LIG AHP onset. Finally, we show that the background climate state modulates precipitation variability with higher variability under wetter background conditions.