We generalize the two-temperature model [Sanders and Walton, Phys. Rev. B, 15, 1489 (1977)] for coupled phonon-magnon diffusion to include the effect of the concurrent magnetization flow. Working within the framework of Boltzmann transport equation, we derive the constitutive equations for coupled phonon-magnon transport driven by gradients of both temperature and external magnetic fields, and the corresponding conservation laws. Our equations reduce to the original Sanders-Walton two-temperature model under a uniform external field, but predict a new magnon cooling effect driven by a non-uniform magnetic field in a homogeneous single-domain ferromagnet. We estimate the magnitude of the cooling effect in yttrium iron garnet, and show it is within current experimental reach. With properly optimized materials, the predicted cooling effect can potentially supplement the conventional magnetocaloric effect in cryogenic applications in the future.Spin caloritronics [1,2] is a nascent field of study that looks into the interaction between heat and spin. In addition to providing ways of thermally manipulating magnetization and magnetic domain walls [3][4][5] as supplements to conventional spintronics, it also holds promise of novel energy harvesting and cooling applications owing to the recent