We use kinetic theory to model the dynamics of a small Bose condensed cloud of heavy particles moving through a larger degenerate Fermi gas of light particles. Varying the Bose-Fermi interaction, we find a crossover between bulk- and surface-dominated regimes - where scattering occurs throughout the Bose cloud or solely on the surface, respectively. We calculate the damping and frequency shift of the dipole mode in a harmonic trap as a function of the magnetic field controlling an interspecies Feshbach resonance. We find excellent agreement between our stochastic model and the experimental studies of Cs-Li mixtures. © 2021 American Physical Society.