We theoretically study the effects of strong magnetoelastic coupling on the transport properties of magnetic insulators. We develop a Boltzmann transport theory for the mixed magnon-phonon modes ("magnon polarons") and determine transport coefficients and the spin diffusion length. Magnon-polaron formation causes anomalous features in the magnetic field and temperature dependence of the spin Seebeck effect when the disorder scattering in the magnetic and elastic subsystems is sufficiently different. Experimental data by Kikkawa [Phys. Rev. Lett. 117, 207203 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.207203] on yttrium iron garnet films can be explained by an acoustic quality that is much better than the magnetic quality of the material. We predict similar anomalous features in the spin and heat conductivity and nonlocal spin transport experiments.