We study droplet dynamics and breakup in generic time-dependent flows via a multicomponent lattice Boltzmann algorithm, with emphasis on flow startup conditions.We first study droplet breakup in a confined oscillatory shear flow via two different protocols. In one setup, we start from an initially spherical droplet and turn on the flow abruptly ("shock method"); in the other protocol, we start from an initially spherical droplet as well, but we progressively increase the amplitude of the flow, by allowing the droplet to relax to the steady state for each increase in amplitude, before increasing the flow amplitude again ("relaxation method"). The two protocols are shown to produce substantially different breakup scenarios. The mismatch between these two protocols is also studied for variations in the flow topology, the degree of confinement, and the inertia of the fluid. All results point to the fact that under extreme conditions of confinement the relaxation protocols can drive the droplets into metastable states, which break only for very intense flow amplitudes, but their stability is prone to external perturbations, such as an oscillatory driving force.