This paper presents calculations of the transport in a Vlasov plasma due to weak electrostatic microfluctuations driven unstable by a shear in a bulk flow. The flow velocity is parallel to a uniform magnetic field and wave propagation is in the plane perpendicular to the shear direction. A consistent procedure is used to evaluate transport of the momentum and the kinetic energy tensor due to the electrostatic flow shear instability. The transport reduces the free energy driving the instability. The wave-particle momentum loss rate scales as the cube of the inverse shear scale length, suggesting that it may be difficult to attain an asymptotically large Reynolds number for macroscopic turbulence in a collisionless plasma.