We observe that an rf microwave field strongly influences the transport of incoherent thermal magnons in yttrium iron garnet. Ferromagnetic resonance in the nonlinear regime suppresses thermal magnon transport by 95%. The transport is also modulated at nonresonant conditions in two cases, both related to the magnon band minimum. Firstly, a strong enhancement of the nonlocal signal appears at a static magnetic field below the resonance condition. This increase only occurs at one field polarity and can be as large as 800%. We attribute this effect to magnon kinetic processes, which give rise to band-minimum magnons and high-energy chiral surface modes. Secondly, the signal increases at a static field above the resonance condition, where the r...