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Modeling mass flows is classically based on hydrostatic balance equations. However, if momentum transfers scale similarly in slope parallel and flow depth directions, then the gravity and acceleration can have the same order of magnitude effects, urging for a non-hydrostatic model formulation. Here, we extend existing single-phase Boussinesq-type models by developing a new non-hydrostatic model for multi-phase mass flows consisting of solid and fine-solid particles and viscous fluid (Pudasaini & Mergili, 2019). The new model includes enhanced gravity and dispersion considering various interfacial momentum transfers. We outline new contributions in the non-hydrostatic Boussinesq-type multi-phase gravity waves emerging from phase-interactions. We present a general, well-structured framework of multi-phase mass flows with enhanced gravity and dispersion, setting a foundation for comprehensive simulations. We discuss situations where non-hydrostatic, dispersive effects are more pronounced for such flows. Reduced models demonstrate the importance of non-hydrostatic contributions. Simplified analytical solutions reveal how the new dispersive model can be reduced to non-dispersive ones, yet largely generalizing existing models. We postulate a novel, spatially varying dissipative force, called the prime-force, which physically controls the dynamics and run-out of the mass flow in a precise way. Practitioners and engineers may find this force very useful in technical applications. This illuminates the need of formally including the prime-force in momentum equations. A simple dispersion equation is derived highlighting the essence of dispersion on mass flow dynamics. Dispersion produces a wavy velocity field about a reference state without dispersion, the first of this kind for avalanching debris mass. Dispersion intensity increases energetically as the solid volume fraction or friction decreases.