The driven dynamics of localization transitions in a non-Hermitian disorder-Aubry-André (DAA) model are examined under both open boundary conditions (OBC) and periodic boundary conditions (PBC). Through an analysis of the static properties of observables, including the localization length ($\xi$), inverse participation ratio ($\mathrm{IPR}$), and energy gap ($\mathrm{\Delta }E$), we found that the critical exponents examined under PBC are also applicable under OBC. The Kibble-Zurek scaling (KZS) for the driven dynamics in the non-Hermitian DAA systems is formulated and numerically verified for different local-to-local quench directions. The hybrid KZS (HKZS) in the overlapping critical region of non-Hermitian DAA and Anderson localization is proposed and numerically confirms the validity across a local-to-skin quench direction. This study generalizes the application of the KZS to the dynamical localization transitions within systems featuring dual localization mechanisms.

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• Critical phenomena
• Dynamical phase transitions
• Phase transitions
• Quantum phase transitions