Dynamical systems subject to perturbations that decay over time are relevant in the description of many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, as well as in celestial mechanics. For this reason, we consider a Hamiltonian dynamical system having an invariant torus supporting arbitrary dynamics, and we study its evolution under a perturbation decaying exponentially over time. By applying a strategy based on a refined analysis of the Banach spaces and functionals involved in the resolution of suitable non-linear invariant equations, we show the existence of orbits converging in time to the arbitrary motions associated with the unperturbed system. As a corollary, an analogous statement for time-dependent vector fields on the torus is also obtained. This result extends to the important case of arbitrary Hamiltonian dynamics a previous work of Canadell and de la Llave where only asymptotic quasi-periodic motions were considered.