This manuscript presents a mathematical model and numerical solution to predict vibration responses of nonlocal strain gradient (NLS) perforated viscoelastic nanobeam under dynamic moving loads. The microstructure and size length scales are considered in the frame of NLS theory. The viscoelastic damping of the structure is considered by linear Kelvin Voigt viscoelastic model. Timoshenko and Euler Bernoulli beam theories are developed to consider both thin and thick structure response. The moving load is portrayed by point load and harmonic type. The dynamic equations of motion incorporating viscoelasticity and size dependency effects is derived by using Hamilton principle. A differential quadrature method (DQM) with Chebyshev–Gauss–Lobatto formula is used to discretize the space domain and solve the model numerically. The obtained results are validated and compared with available literature. Numerical studies are conducted to show the influence of the material viscosity parameter, perforation parameters, shear deformation effect, nonlocal strain gradient, moving load velocity, and beams slenderness ratio on the dynamic behavior of viscoelastic perforated nanobeams under moving load. The material viscosity parameter can effectively control the magnitude and stability of the magnification factor profiles. Higher filling ratios result in larger values of the dynamic magnification factor, whereas larger numbers of hole rows lead to smaller values. The proposed procedure is supportive in the analysis and design of perforated viscoelastic NEMS structures under moving load.