As the hydrodynamic theory breaks down with the local density ap- proximation in the fermionic superfluid with spin-polarization, we develop a general formalism of the adiabatic dynamics for the low-lying collective modes in the BEC-BCS crossover, which is exact in the adiabatic limit. This adi- abatic dynamic theory is based on a static density functional theory of the spin-polarized superfluid system, which we derive as a generalization of the conventional density functional theory of superfluid for current experimental interests. A special case where the system is uniform and analytically solv- able is studied in detail. We show that our adiabatic equations of motion are reduced to the hydrodynamic equations of motion within local density approx- imation, which provides a solid microscopic foundation for the well-publicized phenomenological hydrodynamic theory. / text