Flexible and versatile mobile robotic coworkers are becoming an indispensable commodity for helping humans with repetitive or physically demanding work. A key challenge with these systems is respecting the strict safety requirements in shared and collaborative workspaces. This inevitably requires solving their whole-body dynamics to obtain the necessary inertial impact properties. In this article, we present an integrated impact dynamics and safe precollision control framework to address the discussed challenge. We propose a novel modular dynamics approach that provides efficient formulations for reusing the uncoupled subsystem dynamics when evaluating the coupled system. Our approach is generalized for deriving the whole-body impact dynamics of any articulated floating-base robot. Furthermore, it outperforms classical monolithic approaches for computing the dynamics, making it favorable for systems with more than two dynamic subsystems while allowing decentralized computations. Finally, based on the proposed modular and generalized impact dynamics and extending our previous work, we introduce the generalized safe motion unit as a unified safety scheme for floating-base robotic structures with branched manipulation extremities. The proposed concepts are evaluated on an exemplary wheeled mobile manipulator, considering realistic use cases in simulation and real-world experiments. The obtained results validated the efficacy of our framework and developed methods.