In this paper, we present a unified dynamic modeling approach of (elongated body) continuum robots. The robot is modeled as a geometrically exact beam continuously actuated through an active strain law. Once included in the geometric mechanics of locomotion, the approach applies to any hyperredundant or continuous robot that is devoted to manipulation and/or locomotion. Furthermore, by the exploitation of the nature of the resulting model of being a continuous version of the Newton-Euler model of discrete robots, an algorithm is proposed that is capable of computing the internal control torques (and/or forces), as well as the rigid net motions of the robot. In general, this algorithm requires a model of the external forces (responsible for the self-propulsion), but we will see how such a model can be replaced by a kinematic model of a combination of contacts that are related to terrestrial locomotion. Finally, in this case, which we name “kinematic locomotion,” the algorithm is illustrated through many examples directly related to elongated body animals, such as snakes, worms, or caterpillars, and their associated biomimetic artifacts.