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A droplet entering a contraction does not see the same flow conditions throughout its passage. Here, numerical calculations are used to follow this deformation and determine when breakup occurs for three continuous liquids: one Newtonian and two shear-thinning fluids. The incompressible two-phase problem is solved in OpenFOAM with a Volume-of-Fluid formulation and a modified interDyMFoam solver, while local mesh refinement follows the interface. The droplet is Newtonian. The other two liquids are represented by the Bird–Carreau model and have different high-shear viscosities. For each selected streamline, the initial radius is varied until the limiting size between an intact and a broken droplet is located; the corresponding critical capillary number is then evaluated. The calculations show that elongation is strong but short-lived near the contraction, whereas shear persists farther downstream. They also show why shear rate alone can be misleading for a shear-thinning liquid: close to the wall the shear rate may increase while the viscous stress decreases because apparent viscosity has fallen. Two breakup forms are obtained, namely filament fracture and tip streaming with satellite droplets. The critical radius decreases sharply from the inner trajectories toward the wall and changes little thereafter. The three liquids remain separated when the breakup data are compared using shear rate, but this separation is greatly reduced with a scaled downstream shear-stress coordinate. For the cases considered, the breakup limit therefore reflects the combined changes in deformation rate, apparent viscosity, viscous stress, and viscosity ratio.