In the tropics, oxygen isotope signals of past climate change range from less than 1‰ to upwards of 7‰ or more. Regardless of the amplitude, these signals are often interpreted to reflect changes in local rainout. However, stable isotopes in precipitation can carry information about rainout across thousands of kilometers, making it hard to parse the local and non-local effects. Here, we present a framework that links the amplitude of tropical isotope signals to spatial patterns of climate change that cause them. Using three models of varying complexity, we show that the largest signals require coherent hydrologic change across ~1,000 to 8,000 kilometers. This pattern can be explained by the balance of vapor being rained out versus replenished as it moves over space. Within ~1,000 kilometers, upwind changes in rainout are too localized for a large isotope shift to emerge. Beyond ~8,000 kilometers, the rainout signal is overwhelmed by more locally-sourced vapor. We find that rainout in this ~1,000-8,000 km upwind window causes the largest isotope shifts in tropical paleoclimate, even when the isotope composition is strongly correlated with local precipitation amount. Our results indicate that large amplitude isotope signals are reliable tracers of large scale hydrologic change, and their link to local precipitation amount is tenuous.