How precipitation over ocean regions changes with global warming is grounded in a well accepted theory, but there is so far no such theory to describe precipitation changes over land. This paper provides important theoretical, numerical and observational evidence for the sensitivity of orographically-induced precipitation in the tropics to background wind changes. The result that tropical orographic rainfall increases with cross-slope winds at a rate of 20-30% (which is more than twice the rate expected from simple "upslope flow” theory) has important implications for constraining future climate change, providing an avenue for reliable projections of continental precipitation without the need for climate models that explicitly resolve convection.