In this work, we obtain the leading corrections to the jet momentum broadening distribution in a QCD medium arising from the transverse flow of the matter. We first derive the single-particle propagator of a highly energetic parton resumming its multiple interactions with the homogeneous flowing matter, explicitly keeping the leading subeikonal flow terms. Then, we use this propagator to obtain the jet broadening distribution and its leading moments. We show that this distribution becomes anisotropic in the presence of transverse flow, since its odd moments are generally non-zero and proportional to the transverse velocity of the medium. Finally, we evaluate several odd moments, which we compare to the corresponding results at first order in opacity, showing that accounting for multiple in-medium scatterings is essential to describe some observables in dense nuclear matter.