We study the relaxation dynamics of photocarriers in the paramagnetic Mott insulating phase of the half-filled two-band Hubbard model. Using nonequilibrium dynamical mean-field theory, we excite charge carriers across the Mott gap by a short hopping modulation, and simulate the evolution of the photodoped population within the Hubbard bands. We observe an ultrafast charge-carrier relaxation driven by the emission of local spin excitations with an inverse relaxation time proportional to the Hund's coupling. The photodoping generates additional side-bands in the spectral function, and for strong Hund's coupling, the photodoped population also splits into several resonances. The dynamics of the local many-body states reveals two effects, thermal blocking and kinetic freezing, which manifest themselves when the Hund's coupling becomes of the order of the temperature or the bandwidth, respectively. These effects, which are absent in the single-band Hubbard model, should be relevant for the interpretation of experiments on correlated materials with multiple active orbitals. In particular, the features revealed in the nonequilibrium energy distribution of the photocarriers are experimentally accessible, and provide information on the role of the Hund's coupling in these materials.