Self-sustained smouldering combustion can be a fire hazard, but also a valuable waste-to-energy tool and an effective means for the destruction of organic contaminants. In all contexts, smouldering is an oxygen-limited phenomenon and therefore oxygen mass flux plays a major role. In this study, a multidimensional, thermodynamic-based smouldering model was developed and validated against experiments to quantify the complex interplay between chemical reactions and heat and mass transfer processes. Oxygen supply was independently varied by diluting oxygen mass fraction feeding the smouldering reactions. Smouldering robustness was quantified by local and global energy analyses establishing when a negative net energy balance indicated the onset of quenching. It was confirmed that a diluted oxygen mass fraction resulted in increased heat transfer efficiency driving the smouldering front towards quasi-super-adiabatic conditions. The centreline peak temperature remained almost constant despite a decreasing smouldering front velocity and the weakening local energy balance. Under these unique conditions, key multidimensional heat and mass transfer effects could be explored systematically, showing the displacement of air towards the periphery of the reactor leading to lower peak temperatures and eventually localized quenching. The visual manifestation of localized quenching was an unburnt crust near the reactor wall.