In this letter, the progressive nature of the forming process step in HfO₂-based resistive switching structures is investigated. Contrary to what happens with ramped or pulsed voltage stresses, current-driven degradation experiments shed light on the formation dynamics of the filamentary path across the oxide layer. The resulting voltage-current characteristics are interpreted in terms of electron transport through a mesoscopic constriction with adiabatic shape. The voltage decrease during the forming process is ascribed to a relaxation of the electron wavefunction confinement effect. The role of the compliance level on the leakage current magnitude is also discussed within this framework.