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Global surface warming projections have been empirically connected to carbon emissions via a climate index defined as the transient climate response to emissions (TCRE), revealing that surface warming is nearly proportional to carbon emissions. Here, we provide a theoretical framework to understand the TCRE including the effects of all radiative forcing in terms of the product of three terms: the dependence of surface warming on radiative forcing, the fractional radiative forcing contribution from atmospheric CO₂ and the dependence of radiative forcing from atmospheric CO₂ on cumulative carbon emissions. This framework is used to interpret the climate response over the next century for two Earth System Models of differing complexity, both containing a representation of the carbon cycle: an Earth System Model of Intermediate Complexity, configured as an idealised coupled atmosphere and ocean, and an Earth System Model, based on an atmosphere–ocean general circulation model and including non-CO₂ radiative forcing and a land carbon cycle. Both Earth System Models simulate only a slight decrease in the TCRE over 2005–2100. This limited change in the TCRE is due to the ocean and terrestrial system acting to sequester both heat and carbon: carbon uptake acts to decrease the dependence of radiative forcing from CO₂ on carbon emissions, which is partly compensated by changes in ocean heat uptake acting to increase the dependence of surface warming on radiative forcing. On decadal timescales, there are larger changes in the TCRE due to changes in ocean heat uptake and changes in non-CO₂ radiative forcing, as represented by decadal changes in the dependences of surface warming on radiative forcing and the fractional radiative forcing contribution from atmospheric CO₂. Our framework may be used to interpret the response of different climate models and used to provide traceability between climate models of differing complexity.