We consider the dynamics of an isolated kink band within an otherwise well-aligned lamellar block copolymer (or other smectic A liquid crystal) subjected to a macroscopic shear flow. We find on geometrical grounds that normal relative motion of the tilt boundaries that delineate such a band, relative to the normal velocity of the fluid, can occur only if there is jump in the tangential component of the fluid velocity across the boundary. We show that such tangential slippage should be negligible for well-developed bands with narrow boundaries. These observations lead to a simple description of the evolution of an idealized kink band, in which the kink bandwidth remains constant after the formation of narrow tilt boundaries, and the tilt boundaries rotate as material surfaces in a shear flow. The resulting prediction for the rate of rotation of the layers within such a band is confirmed by in situ small-angle X-ray scattering (SAXS)-steady shear experiments, which measure the evolution of the distribution of lamellar orientations within kink bands in a predominantly parallel poly(styrene-co-ethylenepropylene) diblock copolymer.