We study the coupled transport of phonons and magnons in a single-domain ferromagnet under concurrent temperature and external magnetic field gradients, and show it is a close analogue to the conventional coupled electron-phonon transport that leads to the thermoelectric effect. Working within the framework of Boltzmann transport equation, we derive the constitutive equations for the coupled phonon-magnon transport, and the corresponding conservation laws. Especially we show both the similarity and the difference between the roles played by the electrochemical potential for electrons and the external magnetic field for magnons. 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, which resembles the conventional Peltier effect. 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. This work is supported partially by S3TEC, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences, and partially by the Air Force Office of Scientific Research Multidisciplinary Research Program of the University Research Initiative (AFOSR MURI) via Ohio State University.