A temperature-dependent approach involving Green-Kubo equilibrium atomic and spin dynamics (GKEASD) is reported to assess phonon and magnon thermal transport processes accounting for phonon-magnon interactions. Using body-centered cubic (BCC) iron as a case study, GKEASD successfully reproduces its characteristic temperature-dependent spin and lattice thermal conductivities. The nonelectronic thermal conductivity, i.e., the sum of phonon and magnon thermal conductivities, calculated using GKEASD for BCC Fe, agrees well with experimental measurements. Spectral energy analysis reveals that high-frequency phonon-magnon scattering rates are one order of magnitude larger than those at low frequencies due to energy scattering conservation rules and high densities of states. Higher temperatures further accentuate this phenomenon. This new framework fills existing gaps in simulating thermal transport across the ferro- to paramagnetic transition. Future application of this methodology to phonon- and magnon-dominant insulators and semiconductors will enhance understanding of emerging thermoelectric, spin caloritronic, and superconducting materials.