We present a droplet microfluidic platform mixing the contents of the droplet chaotically in microfluidic induction time measurements, a promising method for quantifying nucleation kinetics with minute amounts of solute. The nucleation kinetics of aqueous potassium chloride droplets dispersed in mineral oil without surfactants is quantified in the presence and absence of chaotic mixing. We demonstrate the ability of the proposed platform to dictate droplet size, to provide a homogeneous temperature distribution, and to chaotically mix the droplet contents. Chaotic mixing in induction time measurements is facilitated by the motion of droplets through serpentine micromixer bends, while the extent of mixing is controlled by how much droplets move. Different nucleation kinetics are observed in experiments where the droplets are static, mixed, and in motion. We hypothesize that the droplet motion induces formation of a thin-liquid Bretherton film surrounding the droplets. The thin film shields droplets from solid boundaries that are more efficient heteronucleant surfaces compared to liquid-liquid interfaces. We observed that repeated microfluidic induction time measurements, particularly with moving droplets, produce significantly distinct cumulative nucleation probability curves, indicating that the measured nucleation kinetics depend strongly on the details of the experimental procedure, which we discuss in detail. Finally, we compare the microfluidic experiments to well-mixed, milliliter volume, turbidity-based measurements in the context of classic nucleation theory.