Biological tissues such as tumors consist of multiple interacting constituents that exchange mass while undergoing growth-induced deformation. To describe these processes in a unified manner, we develop a thermodynamically consistent framework for growing mixtures composed of an arbitrary number of solid and fluid phases. The formulation is posed entirely in the Lagrangian reference configuration and employs the classical multiplicative decomposition of the deformation gradient of a solid skeleton, enabling a clear separation between growth and elastic response, as well as inter-phase mass exchange and transport. Starting from the constituent-wise balances of mass and momentum, we formulate an energy-dissipation law that restricts admissible constitutive relations. This provides a systematic basis for constructing generalized constitutive laws for the free-energy density, growth evolution, and inter-phase mass transfer, such that each process is governed by its conjugate thermodynamic driving force. The resulting governing equations form a coupled system for the phase fields, chemical potentials, and deformation that is consistent with the second law of thermodynamics and applicable to a broad class of growth phenomena. As an application, we specialize the framework to avascular tumor growth. The tumor is modeled as a fully saturated five-phase mixture consisting of multiple solid cell populations interacting with interstitial fluids. Suitable choices of energy densities and source terms enable the description of cell proliferation, competition between cell species, and nutrient-dependent growth. The capabilities of our model are demonstrated through prototypical benchmark problems comparable with studies available in the literature.