The breakup dynamics in non-Newtonian multiphase microsystems is associated with a variety of industrial applications such as food production and biomedical engineering. In this study, we numerically and experimentally characterize the dripping-to-jetting transition under various flow conditions in a Newtonian/shear-thinning multiphase microsystem. Our work can help to predict the formation of undesirable satellite droplets, which is one of the challenges in dispensing non-Newtonian fluids. We also demonstrate the variations in breakup dynamics between shear-thinning and Newtonian fluids under the same flow conditions. For shear-thinning fluids, the droplet size increases when the capillary number is smaller than a critical value, while it decreases when the capillary number is beyond the critical value. The variations highlight the importance of rheological effects in flows with a non-Newtonian fluid. The viscosity of shear-thinning fluids significantly affects the control over the droplet size, therefore necessitating the manipulation of the shear rate through adjusting the flow rate and the dimensions of the nozzle. Consequently, the droplet size can be tuned in a controlled manner. Our findings can guide the design of novel microdevices for generating droplets of shear-thinning fluids with a predetermined droplet size. This enhances the ability to fabricate functional particles using an emulsion-templated approach. Moreover, elastic effects are also investigated experimentally using a model shear-thinning fluid that also exhibits elastic behaviors: droplets are increasingly deformed with increasing elasticity of the continuous phase. The overall understanding in the model multiphase microsystem will facilitate the use of a droplet-based approach for non-Newtonian multiphase applications ranging from energy to biomedical sciences.