A fundamental quantity in tropical dynamics is the ‘convective mass flux’, which measures the upwards mass flux in tropical convection. Changes in convective mass flux under warming are an important, and uncertain, aspect of climate change. Here we build off recent work linking changes in mass flux to the clear-sky energy budget to show that convective mass fluxes decrease along isotherms at around 3-5 % K^-1 under warming. We show that this constraint holds throughout the free-troposphere and across a hierarchy of models; from idealized radiative-convective equilibrium simulations to CMIP6 models. This decrease in convective mass flux with warming is driven by a stabilization of the lapse rate and can be captured with a simple analytical model. We also revisit previous work by Held and Soden (2006), who proposed a scaling for changes in the convective mass flux at cloud-base with warming. We show that the Held and Soden scaling does not capture inter-model spread in cloud-base mass flux changes under warming in cloud-resolving models. Altogether, this work provides a quantitative constraint on changes in convective mass flux throughout the troposphere which can be derived from first principles, and which is verified across a hierarchy of models.