  We calculate the leading corrections to jet momentum broadening and
medium-induced branching that arise from the velocity of the moving medium at
first order in opacity. These results advance our knowledge of jet quenching
and demonstrate how it couples to collective flow of the quark-gluon plasma in
heavy-ion collisions and to the orbital motion of partons in cold nuclear
matter in deep inelastic scattering at the electron-ion collider. We also
compute the leading corrections to jet momentum broadening due to transverse
gradients of temperature and density. We find that these effects lead to both
anisotropic transverse momentum diffusion proportional to the medium velocity
and anisotropic medium-induced radiation emitted preferentially in the
direction of the flow. We isolate the relevant sub-eikonal corrections by
working with jets composed of scalar particles with arbitrary color factors
interacting with the medium by scalar QCD. Appropriate substitution of the
color factors and light-front wave functions allow us to immediately apply the
results to a range of processes including $q \rightarrow q g$ branching in real
QCD. The resulting general expressions can be directly coupled to hydrodynamic
simulations on an event-by-event basis to study the correlations between jet
quenching and the dynamics of various forms of nuclear matter.
