Based on the global interpolating generalized moving least square (IGMLS), an element-free Galerkin (EFG) method for rigid-flexible coupling dynamics of axially moving hub-beam systems is implemented. With the consideration of the geometry nonlinear terms and the longitudinal shrinking induced by the transverse deformation, the firs-order approximate coupling (FOAC) model which both the dynamic stiffening and softening effects can be reflected is built. The global interpolating and element-free properties makes some advantages of the EFG: the convenient imposition for both displacement and derivative boundary conditions and the direct obtaining of actual nodal values compared with the standard EFG method; convenience in dealing with the rigid-flexible coupling terms, constructing highly continuous shape functions and assembling the whole spatial discretized equations by comparing to FEM. Numerical comparisons between the EFG and analytical solutions on the vibration of a cantilever beam show the good accuracy and efficiency of the EFG based on the proper option of related numerical parameters. Numerical results of the axially moving hub-beam system demonstrate the reasonability of FOAC model and the feasibility of the global IGMLS applied for rigid-flexible coupling dynamics.