We present the design of a concentric tube (CT) reactor for roll-to-roll chemical vapor deposition (CVD) on flexible substrates, and its application to continuous production of graphene on copper foil. In the CTCVD reactor, the thin foil substrate is helically wrapped around the inner tube, and translates through the gap between the concentric tubes. We use a bench-scale prototype machine to synthesize graphene on copper substrates at translation speeds varying from 25 mm/min to 500 mm/min, and investigate the influence of process parameters on the uniformity and coverage of graphene on a continuously moving foil. At lower speeds, high-quality monolayer graphene is formed; at higher speeds, rapid nucleation of small graphene domains is observed, yet coalescence is prevented by the limited residence time in the CTCVD system. We show that a smooth isothermal transition between the reducing and carbon-containing atmospheres, enabled by injection of the carbon feedstock via radial holes in the inner tube, is essential to high-quality roll-to-roll graphene CVD. We discuss how the foil quality and microstructure limit the uniformity of graphene over macroscopic dimensions. We conclude by discussing means of scaling and reconfiguring the CTCVD design based on general requirements for 2-D materials manufacturing.