We examine the tropical inversion strength, measured by the Estimated Inversion Strength (EIS), and its response to climate change in 18 models associated with phase 5 of the Coupled Model Intercomparison Project (CMIP5). While CMIP5 models generally capture the geographic distribution of observed EIS, they systematically underestimate it off the west coasts of continents, due to a warm bias in sea surface temperature. The negative EIS bias may contribute to the low bias in tropical low-cloud cover in the same models. Averaged over all models, the areal coverage of inversion increases by 10% (in absolute terms) over the 21st century under Representative Concentration Pathway 8.5, and the mean inversion strength increases by 0.8 K. Climate simulations for the Last Glacial Maximum show systematic decreases in the inversion coverage and strength, suggesting that tropical inversion changes are symmetric with respect to the sign of climate change. These inversion changes can be broken down into the fast response to anthropogenic forcing, which is strongly impacted by nearly instantaneous continental warming, and the temperature-mediated component. The latter can be broken down further into contributions from uniform and non-uniform oceanic warming. Surprisingly, substantial EIS increases are found in uniform oceanic warming simulations, due to warming with height exceeding the moist adiabatic rate (though the reasons for this enhancement are still unclear). Nonuniform oceanic warming also plays a large role in EIS change. Finally, we show that future EIS change is exaggerated by about 20\% when one assumes a constant surface relative humidity in calculating EIS as in its original definition.