The temperature dependences of the specific heat and the transport characteristics of thermal-frequency phonons in single crystals of the solid solutions of rare-earth aluminum garnets are studied at liquid-helium temperatures in the presence of Schottky-type low-energy excitations. The kinetic characteristics of phonons as functions of the solid solution composition are measured. The relation between the kinetic and thermophysical characteristics of the material in the solid solutions of rare-earth aluminum garnets is analyzed under conditions of a nonstationary process and the spatial inhomogeneity caused by the coordinate dependence of the state of low-energy excitations. The thermalization conditions in the nonequilibrium phonon-low-energy excitation system are estimated.