Dynamic nuclear polarization (DNP) and emerging quantum technologies rely on the spin transfer in electron-nuclear hybrid quantum systems. Spin transfers might be suppressed by larger couplings, e.g., hyperfine couplings suppressing nuclear dipolar flip-flops ("spin diffusion barrier"). We apply the Schrieffer-Wolff transformation to a two-electron two-nucleus spin system involving dipolar and hyperfine couplings in their tensorial form and study possible polarization-transfer processes. Among the different effective Hamiltonian matrix elements investigated is an energy-conserving electron-nuclear four-spin flip-flop, which combines an electronic with a nuclear dipolar flip-flop. The relevance of this electron-nuclear four-spin flip-flop for nuclear spin diffusion close to electrons is supported by model fits of HypRes-on experimental data. We connect the closely related fields of magnetic resonance and quantum information and provide a model that explains how all nuclear spins can contribute to the hyperpolarization of the bulk without a spin diffusion barrier in DNP.