Shifts in the latitude of the intertropical convergence zone - a region of intense tropical rainfall - have often been explained through changes in the atmospheric energy budget, specifically through theories that tie rainfall to meridional energy fluxes. These quantitative theories can explain shifts in the zonal mean, but often have limited relevance for regional climate shifts, such as a period of enhanced precipitation over Saharan Africa during the mid-Holocene. Here we present a theory for regional tropical rainfall shifts that utilizes both zonal and meridional energy fluxes. We first identify a qualitative link between zonal and meridional energy fluxes and rainfall variations associated with the seasonal cycle and the El Ni{ñ}o/Southern Oscillation. We then develop a quantitative theory based on these fluxes that relates atmospheric energy transport to tropical rainfall. When applied to the orbital configuration of the mid-Holocene, our theory predicts continental rainfall shifts over Africa and Southeast Asia that are consistent with complex model simulations. However, the predicted rainfall over the Sahara is not sufficient to sustain vegetation at a level seen in the palaeo-record, which instead requires an additional large energy source such as that due to reductions in Saharan surface albedo. We thus conclude that additional feedbacks, such as those involving changes in vegetation or soil type, are required to explain changes in rainfall over Africa during the mid-Holocene.