A key step in nutrient sensing is activation of the master growth regulator, mTORC1 kinase, on the lysosomal membrane. Nutrients enable mTORC1 scaffolding by a complex composed of the Rag GTPases (Rags) and Ragulator, but the underlying mechanism of mTORC1 capture is poorly understood. Combining dynamic imaging in cells and reconstituted systems, we uncover an affinity switch that controls mTORC1 lifetime and activation at the lysosome. Nutrients destabilize the Rag-Ragulator interface, causing cycling of the Rags between lysosome-bound Ragulator and the cytoplasm, and rendering mTORC1 capture contingent on simultaneous engagement of two Rag-binding interfaces. Rag GTPase domains trigger cycling by coordinately weakening binding of the C-terminal domains to Ragulator in a nucleotide-controlled manner. Cancer-specific Rag mutants override release from Ragulator and enhance mTORC1 recruitment and signalling output. Cycling in the active state sets the Rags apart from most signalling GTPases, and provides a mechanism to attenuate mTORC1 signalling.