Aqueous polymer colloids known as latexes are widely used in coating applications. Multicomponent latexes comprised of two incompatible polymeric species organized into a core-shell particle morphology are a promising system for self-stratifying coatings that spontaneously partition into multiple layers, thereby yielding complex structured coatings requiring only a single application step. Developing new materials for self-stratifying coatings requires a clear understanding of the thermodynamic and kinetic properties governing phase separation and polymeric species transport. In this work, we study phase separation and self-stratification in polymer films based on multicomponent acrylic (shell) and acrylic-silicone (core) latex particles. Our results show that the molecular weight of the shell polymer and heat aging conditions of the film critically determine the underlying transport phenomena, which ultimately controls phase separation in the film. Unentangled shell polymers result in efficient phase separation within hours with heat aging at reasonable temperatures, whereas entangled shell polymers effectively inhibit phase separation even under extensive heat aging conditions over a period of months due to kinetic limitations. Transmission electron microscopy is used to track morphological changes as a function of thermal aging. Interestingly, our results show that the rheological properties of the latex films are highly sensitive to morphology, and linear shear rheology is used to understand morphological changes. Overall, these results highlight the importance of bulk rheology as a simple and effective tool for understanding changes in morphology in multicomponent latex films.