We propose a novel message-passing solution to the sixth-dimensional (6D) moving rigid body localization (RBL) problem, in which the three-dimensional (3D) translation vector and rotation angles, as well as their corresponding translational and angular velocities, are all estimated by only utilizing the relative range and Doppler measurements between the "anchor" sensors located at an 3D (rigid body) observer and the "target" sensors of another rigid body. The proposed method is based on a bilinear Gaussian belief propagation (GaBP) framework, employed to estimate the absolute sensor positions and velocities using a range- and Doppler-based received signal model, which is then utilized in the reconstruction of the RBL transformation model, linearized under a small-angle approximation. The method further incorporates a second bivariate GaBP designed to directly estimate the 3D rotation angles and translation vectors, including an interference cancellation (IC) refinement stage to improve the angle estimation performance, followed by the estimation of the angular and the translational velocities. The effectiveness of the proposed method is verified via simulations, which confirms its improved performance compared to equivalent state-of-the-art (SotA) techniques.