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Large-scale peculiar motions are commonplace in our universe. Nevertheless, their origin, evolution and implications are still largely unknown. It is generally assumed that bulk motions are a relatively recent addition to the universal kinematics, triggered by the increasing inhomogeneity and anisotropy of the post-recombination epoch. In this work, we focus on the linear evolution of peculiar velocities prior to recombination, namely in the late radiation era and also during a phase of de Sitter inflation. We begin by showing/confirming that bulk motions are triggered and sustained by the non-gravitational forces developed during structure formation. Since density and therefore peculiar-velocity perturbations cannot grow in the baryonic sector before recombination, we consider drift motions in non-baryonic species, which can start growing in the late radiation era. Using relativistic linear cosmological perturbation theory, we find that peculiar motions in the low-energy dark component exhibit power-law growth, which increases further after equipartition. Turning to the very early universe, we consider the evolution of linear peculiar velocities during a phase of de Sitter inflation. We find that typical slow-roll scenarios do not source peculiar motions. Moreover, even if the latter were to be present at the onset of the de Sitter phase, the subsequent exponential expansion should quickly wash away any traces of peculiar-velocity perturbations., Comment: Typos corrected. Published version