To analyze the dynamic response of soft core sandwich beams under a moving point load, an N-node weak form quadrature sandwich beam element is established based on the extended high order sandwich panel theory and Hamilton principle. The formulas to calculate the element stiffness and mass matrices are given using the explicit formulas of the weighting coefficients existing in the differential quadrature method. The correctness and efficiency of the element stiffness and mass matrices are verified. It is shown that the weak form quadrature element method has advantages of high accuracy and computational efficiency. The dynamic response of a soft core sandwich beam with both ends clamped and subjected to a moving point load is obtained by using the finite difference method for the first time. Present research extends the application range of the weak form quadrature element method.