This study investigates the dynamic behavior of a nanoscale beam subjected to a moving load within a thermal setting. A new spatiotemporal nonlocal elasticity framework is introduced, incorporating a dynamic scalar nonlocal kernel. Based on this framework, an isotropic nonlocal elasticity model of the Klein–Gordon form is developed, integrating both an intrinsic length scale and a crucial time scale parameter. Heat transport in this problem, driven by an exponentially decaying thermal field, is governed by the refined Moore–Gibson–Thompson (MGT) theory, which includes a higher-order memory-dependent derivative (MDD). The nanobeam is modeled with simple supports at its axial extremities. Solutions for the governing equations are derived using the Laplace transform technique. Numerical evaluations of the physical fields are conducted to analyze the impact of spatiotemporal nonlocality parameters, moving load velocity, and thermal decay time. Specific case studies are presented to validate the proposed model.