This work presents the first covariant cosmological formulation of the RMB (Raum–Materie–Bewegung) theory, in which motion acts as a geometric source of stress–energy. We construct the unique FRW-compatible RMB vector field Φμ = f(H) uμ and introduce a dynamical screening mechanism that suppresses the motion-induced stress term at early times while activating it near the present epoch. A complete derivation of the modified Friedmann equations is given, including energy conservation, background stability, and the identification of a late-time attractor solution. The model naturally yields a mild enhancement of the late-time expansion rate H(z), offering a plausible mechanism that alleviates (but does not fully resolve) the Hubble tension without invoking dark energy or new scalar fields. Four appendices provide the technical details: (A) tensorial derivation of the modified Friedmann equations, (B) energy conservation in an expanding background, (C) a dynamical-system analysis ensuring stability and the absence of divergences, and (D) implementation notes and numerical considerations. A representative H(z) comparison plot with ΛCDM is included. This preprint establishes the theoretical groundwork for RMB Cosmology II, which will extend the framework to linear perturbations, structure formation, and observational data (BAO, SNe, CMB).