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This paper addresses the problem of steering a robotic vehicle along a geometric path specified with respect to a reference frame moving in three dimensions, termed the Moving Path Following (MPF) motion control problem. The MPF motion control problem is solved for a large class of robotic vehicles that require a minimum positive forward speed to operate, which poses additional constraints, and is developed using geometric concepts, wherein the attitude control problem is formulated on Special Orthogonal group SO(3). Furthermore, the proposed control law is derived from a novel MPF error model formulation that allows to exclude the conservative constraints on the initial position of the vehicle with respect to the reference path by enabling the explicit control of the progression of a virtual point moving along the reference path. The task of the MPF control law is then to steer the vehicle towards the moving path and converge to the virtual point. Formal stability and convergence guarantees are provided using the Input-to-State Stability concept. In particular, we show that the proposed controller is robust to imperfect tracking errors by the autopilot and wind gusts. Simulation results are presented to illustrate the efficacy of the proposed MPF control law.