Two Earth system models are analyzed to gain insight into the processes that govern projected changes in the South Asian monsoon. Warmer present-day base state tropical SSTs contribute to coupled processes that produce greater future tropical Pacific warming in CESM2 with less of an increase in season-mean monsoon precipitation compared to E3SMv2. This is attributed to changes in the large-scale east-west atmospheric Walker circulation, with relatively larger increases in precipitation and upper-level divergence over the tropical Pacific and increases in upper-level convergence over South Asia in CESM2. The stronger El Niño-like response in CESM2, which increases Pacific precipitation and upper-level divergence farther to the east, and larger future ENSO amplitude in E3SMv2, produce a greater relative increase in future monsoon-ENSO connections in E3SMv2 compared to CESM2. This analysis indicates that the key processes that affect future monsoon-ENSO connections are ENSO amplitude and size of the future tropical Pacific El Niño-like response.