An atmospheric composition feedback mechanism modulates the global equilibrium climate sensitivity (ECS) through changes in the tropical upper-tropospheric and lower-stratospheric (UTLS) water vapor. The feedback mechanism is caused by the acceleration of the Brewer-Dobson circulation. This process changes the ozone (O$_{3}$) concentration, resulting in a drier and cooler UTLS region than without O$_{3}$ changes. Thus, the planetary long-wave emissivity increases, and the ECS decreases. However, the BDC alone already provides dynamical cooling through the tropical stratospheric upwelling, potentially impacting the ECS. Here, we analyze the implications of this effect from a tropical clear-sky perspective, applying a one-dimensional radiative-convective equilibrium (RCE) model that explicitly accounts for the adiabatic cooling by the BDC and includes an interactive representation of O$_{3}$. We study how increasing upwelling modifies the change of the tropical energy budget resulting from a doubling of CO$_{2}$. An increase in upwelling reduces the tropical ECS mainly through an increased tropical energy export related to the adiabatic cooling. The atmospheric composition feedback through O$_{3}$ contributes less than 30\% to the tropical ECS reduction. Due to the dominance of the energy export, any impact on the global ECS will depend on the redistribution of the energy in the extratropics. We show that GCMs simulate similar changes of the tropical energy export under increased upwelling which corroborates that the findings obtained with the RCE approach bear relevance for the global climate.