It is now widely recognized that cellular motility arises from the polymerization of actin---one of the most abundant proteins in eukaryotic cells---into a dynamic and interconnected filament network. In our continuum mechanics formulation, we describe this process by positing that actin polymerization generates a localized mechanical swelling around nucleation sites, which ultimately drives cellular or even bacterial locomotion. To elucidate the mechanobiology underlying these phenomena, we combine state-of-the-art microscopy with a newly developed theoretical framework, enabling advanced modeling and simulation. Departing from most existing mixture theories, we remove the constraint of constituent incompressibility and extend the classical chemo-transport-mechanics formulation . This enriched theory proves well suited for modeling cell motility and holds promise for broader applications across mechanobiology, such as blood clotting and biofilms formation.