Numerical time integration still constitutes a rather challenging task when applied to quite stiff structural systems, especially in the presence of complex vibration scenarios. In this context, the accurate evaluation of all displacement, velocity and acceleration dynamic response variables, specifically due to a Moving Load (ML) excitation, poses serious difficulties in possibly reconciling calculated and measured dynamic responses. The present work refers to a novel computational strategy allowing for an “exact” integration within the time-step, and benchmarks its outcomes for the comparison with the numerical integration by known methods. The analysis is exemplified on a continuous beam with multiple spans, including ML transit and subsequent window of free vibration response. The broader underlying framework is that of Structural Health Monitoring (SHM) of a historic railway bridge, which truly challenges the reliability of integration methods in correctly tracing the current dynamic response, in view of system identification and digital twin replication, toward informed management and maintenance scenarios.