A common theoretical framework is presented to model the conduction characteristics of the two main dielectric breakdown modes in ultrathin SiO2 gate oxides, namely the soft and hard breakdown modes. The breakdown paths are considered to behave as mesoscopic quantum point contacts so that the conduction properties are controlled by energy funneling effects. An adiabatic approach to the modelling of these quantum point contacts is adopted to obtain an analytical approximation for the total transmission coefficient. In the limit of small breakdown spot areas, tunneling through a potential barrier associated with the lower electron transversal state at the narrowest part of the constriction explains the soft breakdown characteristics. For larger areas, such a barrier dissapears and conductance quantization is predicted. Experimental results are well explained by the model, including the conductance values after the hard breakdown and the oxide thickness independence of the current-voltage characteristic after the soft breakdown.