The progressive leakage current growth in electrically stressed Hf_xSi_1-xON/SiON gate stacks is analyzed within the framework of the physics of mesoscopic conducting systems. The breakdown spot is modeled as a nanoconstriction whose effective cross-section area increases as the degradation proceeds. We show that, after eliminating the gate tunneling current component, the post-breakdown conductance exhibits plateaus close to the quantum unit 2e²/h, where e is the electron charge and h the Planck's constant, as it is expected for atomic-sized contacts.