This article presents a unified dynamic modeling approach of continuum robots. The robot is modeled as a geometrically exact beam continuously actuated through an active strain law. Once included into the geometric mechanics of locomotion, the approach applies to any hyper-redundant or continuous robot devoted to manipulation and/or locomotion. Furthermore, exploiting the nature of the resulting models as being a continuous version of the Newton-Euler models of discrete robots, an algorithm is proposed which is capable of computing the internal control torques (and/or forces) as well as the rigid overall motions of the locomotor robot. The efficiency of the approach is finally illustrated through many examples directly related to the terrestrial locomotion of elongated animals as snakes, worms or caterpillars and their associated bio-mimetic artifacts.