The surface warming response to carbon emissions is diagnosed using a suite of Earth system models, 9 CMIP6 and 7 CMIP5, following an annual 1% rise in atmospheric CO2 over 140 years. This surface warming response defines a climate metric, the Transient Climate Response to cumulative carbon Emissions (TCRE), which is important in estimating how much carbon may be emitted to avoid dangerous climate. The processes controlling these intermodel differences in the TCRE are revealed by defining the TCRE in terms of a product of three dependences: the surface warming dependence on radiative forcing (including the effects of physical climate feedbacks and planetary heat uptake), the radiative forcing dependence on changes in atmospheric carbon and the airborne fraction. Intermodel differences in the TCRE are mainly controlled by the thermal response involving the surface warming dependence on radiative forcing, which arise through large differences in physical climate feedbacks that are only partly compensated by smaller differences in ocean heat uptake. The other contributions to the TCRE from the radiative forcing and carbon responses are of comparable importance to the contribution from the thermal response on timescales of 50 years and longer for our subset of CMIP5 models and 100 years and longer for our subset of CMIP6 models. Hence, providing tighter constraints on how much carbon may be emitted based on the TCRE requires providing tighter bounds for estimates of the physical climate feedbacks, particularly from clouds, as well as to a lesser extent for the other contributions from the rate of ocean heat uptake, and the terrestrial and ocean cycling of carbon.