Climate models project that tropical warming is amplified aloft in response to increased CO$_2$. Amplification aloft is expected following moist adiabatic adjustment and the Clausius-Clapeyron relation. Here, we show that moist adiabatic adjustment overpredicts the multi-model mean temperature response at 300 hPa by 12.9–25.3\% across the model hierarchy. We show that overprediction is influenced by at least three mechanisms: large-scale circulation, direct effect of CO$_2$, and convective entrainment. Accounting for the large-scale circulation and the direct effect of CO$_2$ reduces overprediction by 5.7\% and 3.8\% respectively, but does not eliminate it. To test the influence of entrainment, we vary the Tokioka parameter in aquaplanet simulations with and without a large-scale circulation. When varying the climatological entrainment rate in the aquaplanet, overprediction varies from 6.7–17.9\%. The sensitivity of overprediction to climatological entrainment rate in the aquaplanet configured in radiative-convective equilibrium agrees well with the predictions of zero-buoyancy bulk-plume models.