A motion continuous control scheme for fully damped mechanical systems with constrained inputs is proposed. It gives the freedom to choose among finite-time and (local) exponential convergence through a simple design parameter. The control objective is achieved from any initial conditions, for desired trajectories that can be physically tracked avoiding actuator saturation and loss of motion error dissipation, globally induced through the aid of the natural damping terms explicitly considered in the open-loop dynamics. The stability analysis is based on a strict Lyapunov function and is formally developed within an appropriate analytical framework that takes into account the time-varying character naturally adopted by the closed loop. Simulation tests are further included.