Breakup of small aggregates is governed by the imbalance of imposed hydrodynamic forces and cohesive forces between constituent particles. Aggregate restructuring in ramped shear flows at infinitely low Reynolds number are known to reinforce aggregates, increasing effective cohesive strength. However, non-negligible flow inertia is known to increase breakage rates, and is expected to affect breakage kinetics under finite Reynolds number conditions in accelerated flows. A numerical investigation was conducted to establish the effect of flow acceleration on aggregate evolution. Aggregates were characterized by their size, structure and interparticle forces. Individual aggregates were subjected to accelerating flows imposed through shear stresses at the boundaries, and their structural evolution along with breakage events were recorded. Particles were tracked with Discrete Element Method. The flow was solved using a Lattice Boltzmann method, and two-way coupling between the solid and liquid phase was achieved through an Immersed Boundary Method. The findings show that although aggregates restructure due to the shear flow, their structure at breakage does not depend on shear stress. Increasing flow acceleration is found to slow down aggregate breakage and rotation, despite higher imposed shear stresses at the boundaries of the domain. The observed delays is found to be a transient effect of flow inertia around the aggregates. The reported findings establish a novel addition to the criteria for aggregate breakage, where, along with shear strength of the aggregates, flow accelerations and Reynolds number at the scale of the aggregates must also be considered.