In this paper, an integrated nonclassical multi-physics model is developed to study the coupling effect of surface energy and local microstructure on the nonlinear size-dependent pull-in instability of electrostatically actuated functionally graded material (FGM) micro-/nanoswitches. The developed model incorporates the influences of fringing field, dispersion Casimir or van der Waals force in addition to residual axial stress and mid-plane stretching. For more accurate analysis of the FGM switches, a nonclassical beam model is developed based on the Euler–Bernoulli beam theory in conjunction with the modified couple stress theory and Gurtin–Murdoch surface elasticity theory to account for the size dependency and surface energy effects, respectively. Material properties of both bulk and surface layers of the FGM switch are assumed to vary according to a power law distribution through thickness. To this end, the Hamilton principle is employed to derive the nonlinear size-dependent governing integral–differential equations and the associated nonclassical boundary conditions, without neglecting any terms raised by surface energy. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including surface energy theory, modified couple stress and classical theories. The resulting nonlinear integral–differential equations are solved utilizing the generalized differential/integral quadrature method, in which the nonclassical boundary conditions are exactly implemented for different immovable ends. The obtained results are compared with the results available in the literature to valid the efficiency of the present solution method. A numerical analysis reveals that the nonlinear pull-in voltage of FGM micro-/nanoswitches is significantly influenced by the FGM’s gradient index, length scale parameter, surface energy, residual axial stress, initial gap, slenderness ratio, and dispersion forces for different immovable boundary conditions.