Manipulation and sorting of objects in non-Newtonian fluids in microfluidic confinement have immense importance in biological and industrial applications. However, the migration of droplets in non-Newtonian fluids has been rarely investigated experimentally. Also, the role of shear-thinning in migration dynamics of rigid particles is not well-understood. We study lateral migration of viscoelastic and Newtonian droplets in a viscoelastic matrix at a very low Reynolds number. Depending on viscosity ratio and elasticity ratio, viscoelastic droplets can migrate to either channel wall or center while Newtonian droplets migrate to the center for all viscosity ratios. Interestingly, in strongly shear-thinning viscoelastic fluids, droplets exhibit size-dependent migration, i.e. bigger droplets follow their original streamline while smaller ones migrate to the channel center. These observations pave the path for the size and viscoelasticity-based sorting of droplets in the non-Newtonian medium. In stark contrast to the above observations, rigid microparticles in shear-thinning viscoelastic fluids exhibit five different migration regimes, namely, Original streamline, Bimodal, Center migration, Defocusing, and Wall migration. Our investigations will lead to a better understanding of the migration dynamics of cells and microswimmers in biological and synthetic non-Newtonian fluids.