Tropical convection is expected to decrease with warming, in a variety of ways. Specific incarnations of this idea include the ‘stability-iris’ hypothesis, as well as the decrease of both tropospheric and cloud-base mass fluxes with warming. This paper seeks to encapsulate these phenomena into three ‘rules’, and to explore their interrelationships and robustness, using both analytical reasoning as well as cloud-resolving and global climate simulations. We find that each of these rules can be derived analytically from the usual expression for clear-sky subsidence, so they all embody the same essential physics. But, these rules do not all provide the same degree of constraint: the stability-iris effect is not entirely robust due to relatively unconstrained microphysical degrees of freedom, and similarly the decrease in cloud-base mass flux is not entirely robust due to unconstrained effects of entrainment and detrainment. Tropospheric mass fluxes, on the other hand, are shown to be well-constrained theoretically, and when evaluated in temperature coordinates they exhibit a monotonic decrease with warming, at all levels and across a hierarchy of models.