In this study, a modified finite element method (FEM) that can be used to analyse the transverse vibrations of a Timoshenko beam, made of functionally graded materials (FGMs), on a two-parameter foundation and subjected to a variable-velocity moving mass is presented. First, the motion equations of the FGM beam exposed to a moving mass are obtained by combining the mass interaction equations with the beam equations obtained using the first-order shear deformation theory (FSDT). Secondly, to obtain the equation of motion of the beam, mass and foundations, the rigidity of the two-parameter foundations, the Winkler and shear, then combined with the motion equations of the beam and mass using the Hamilton principle. The response of the system is investigated considering the effects of inertia and variable velocity of the moving mass, parameters of the foundations and material constituents of the FGM beam. The changes on the dynamic behaviour of the FGM Timoshenko beam, which are affected by the frequency variation of the considered system are highlighted.