Tipping elements are nonlinear subsystems of the Earth system that can potentially abruptly and irreversibly shift if environmental change occurs. Among these tipping elements is the Amazon rainforest, which is threatened by anthropogenic activities and increasingly frequent droughts. Here, we assess how extreme deviations from climatological rainfall regimes may cause local forest-savanna transitions that cascade through the coupled forest-climate system. We develop a dynamical network model to uncover the role of atmospheric moisture recycling in such tipping cascades. We account for the heterogeneity in critical thresholds of the forest caused by adaptation to local climatic conditions. Our results reveal that, despite this adaptation, increased dry-season intensity may trigger tipping events particularly in the southeastern Amazon. Moisture recycling is responsible for one-fourth of the tipping events. If the rate of climate change exceeds the adaptive capacity of some parts of the forest, secondary effects through moisture recycling may exceed this capacity in other regions, increasing the overall risk of tipping across the Amazon rainforest.