<p>The dynamic behavior of the sandwich beam with the viscoelastic core and carbon nanotube-reinforced polymer face-sheets under the moving mass is the subject of this paper. This beam has significant applications in bridge design and construction. The analysis utilizes a three-layer sandwich beam theory, which combines Frostig’s theory for the core and Timoshenko beam theory for the face-sheets. The frequency-dependent model is employed for the viscoelastic core, which has the advantage of accounting for the frequency dependence of material properties in viscoelastic materials, as opposed to simpler models. Hamilton’s principle is employed to derive the regulating equations in accordance with this theory. Navier’s analytical method is employed to solve the equilibrium equations in the spatial domain, while the Newmark numerical method is employed in the time domain. In order to verify the precision of the findings, they are compared to existing solutions for particular cases. This study aims to investigate the dynamic response of sandwich beams with a frequency-dependent viscoelastic core and compare it with the response predicted by the Kelvin–Voigt model. Additionally, the study aims to investigate the impact of parameters such as the volume fraction of carbon nanotubes, geometric ratios, and dimensional parameters on the critical velocity and dynamic response of the moving mass. The frequency-dependent model produces a more precise representation of the transient response of the sandwich beam than the Kelvin–Voigt model, as indicated by the results. Furthermore, the critical velocity measured by the frequency-dependent core model is 4% lower than that measured by the Kelvin–Voigt model.</p>