The present paper investigates the dynamic characteristics and stability of moving functionally graded material rectangular thin plate. Based on Voigt model, the material properties are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. By the first order shear deformation theory, the differential equations of motion of the moving FGM rectangular plate are derived. The vibration frequencies are obtained from the solution of a generalized eigenvalue problem. Entire computational work is carried out in a normalized square domain obtained through an appropriate domain mapping technique. Results of the reduced problem revealed excellent agreement with other studies. The dimensionless complex frequencies of the moving FGM rectangular plate with four edges simply supported are calculated by the differential quadrature method. The effects of gradient index, aspect ratio, and dimensionless moving speed on the transverse vibration and stability of the moving FGM rectangular plate are analyzed. Results are furnished in dimensionless amplitude–frequency vs. dimensionless moving speed in the form of curves and pictorial representations of some vibration mode shapes are made.