Plants continuously extend their root and shoot systems through the action of meristems at their growing tips. By regulating which meristems are active, plants adjust their body plans to suit local environmental conditions. The transport network of the phytohormone auxin has been proposed to mediate this systemic growth coordination, due to its self-organising, environmentally sensitive properties. In particular, a positive feedback mechanism termed auxin transport canalization, which establishes auxin flow from active shoot meristems (auxin sources) to the roots (auxin sinks), has been proposed to mediate competition between shoot meristems and to balance shoot and root growth. Here we provide strong support for this hypothesis by demonstrating that a second hormone, strigolactone, regulates growth redistribution in the shoot by rapidly modulating auxin transport. A computational model in which strigolactone action is represented as an increase in the rate of removal of the auxin export protein, PIN1, from the plasma membrane can reproduce both the auxin transport and shoot branching phenotypes observed in various mutant combinations and strigolactone treatments, including the counterintuitive ability of strigolactones either to promote or inhibit shoot branching, depending on the auxin transport status of the plant. Consistent with this predicted mode of action, strigolactone signalling was found to trigger PIN1 depletion from the plasma membrane of xylem parenchyma cells in the stem. This effect could be detected within 10 minutes of strigolactone treatment and was independent of protein synthesis but dependent on clathrin-mediated membrane trafficking. Together these results support the hypothesis that growth across the plant shoot system is balanced by competition between shoot apices for a common auxin transport path to the root and that strigolactones regulate shoot branching by modulating this competition.