Fluid injection is widely used to enhance permeability in rock formations by creating or dilating transport pathways for resources such as oil, gas, heat, or CO2. The dynamic propagation of damage induced by fluid injection is governed by fluid flow, dynamic poroelastic deformation, mixed tensile and shear failure, and damage-induced antipermeability degradation. However, the transition from elastoplastic deformation to mixed-mode failure, as well as the induced dynamics, remains ambiguous. This study combines the dynamic Biot's poroelasticity and coupled Drucker-Prager plasticity, Grady-Kipp damage, and antipermeability degradation to simulate dynamic hydraulic fracturing. An explicit predictor-corrector scheme was employed to solve the dynamics of saturated porous media and identify the key factors controlling dynamic damage propagation. The proposed model was tested on soil column consolidation and rock hydraulic fracturing driven by a pre-existing crack, demonstrating good agreement between the numerical and experimental results. Simulation results indicate that damage zones facilitate preferential flow during fluid injection due to damage-induced degradation. The most extensive damage zone is observed under strong damage-permeability coupling. Shear plasticity, tensile damage, and induced seismicity are dominated by fracturing dynamics induced by fluid injection. Oscillations in the temporal-spatial evolution of damaged and plastic points, cumulated potency, and moment magnitude confirm the fracturing dynamics. Shorter injection times result in stronger dynamics and more significant damage propagation. The period of oscillations in cumulated potency increases with injection time while their amplitude gradually decreases due to energy release. These findings highlight injection-induced fracturing dynamics, offering novel insights into the dynamic propagation of damage coupled with matrix antipermeability degradation.