In a previous study the quasi-instantaneous chemical impacts (rapid adjustments) of strongly enhanced methane (CH4) mixing ratios have been analysed. However, to quantify the influence of the respective slow climate feedbacks on the chemical composition it is necessary to include the radiation-driven temperature feedback. Therefore, we perform sensitivity simulations with doubled and quintupled present-day (year 2010) CH4 mixing ratios with the chemistry–climate model EMAC (European Centre for Medium-Range Weather Forecasts, Hamburg version – Modular Earth Submodel System (ECHAM/MESSy) Atmospheric Chemistry) and include in a novel set-up a mixed-layer ocean model to account for tropospheric warming. Strong increases in CH4 lead to a reduction in the hydroxyl radical in the troposphere, thereby extending the CH4 lifetime. Slow climate feedbacks counteract this reduction in the hydroxyl radical through increases in tropospheric water vapour and ozone, thereby dampening the extension of CH4 lifetime in comparison with the quasi-instantaneous response. Changes in the stratospheric circulation evolve clearly with the warming of the troposphere. The Brewer–Dobson circulation strengthens, affecting the response of trace gases, such as ozone, water vapour and CH4 in the stratosphere, and also causing stratospheric temperature changes. In the middle and upper stratosphere, the increase in stratospheric water vapour is reduced with respect to the quasi-instantaneous response. We find that this difference cannot be explained by the response of the cold point and the associated water vapour entry values but by a weaker strengthening of the in situ source of water vapour through CH4 oxidation. However, in the lower stratosphere water vapour increases more strongly when tropospheric warming is accounted for, enlarging its overall radiative impact. The response of the stratosphere adjusted temperatures driven by slow climate feedbacks is dominated by these increases in stratospheric water vapour as well as strongly decreased ozone mixing ratios above the tropical tropopause, which result from enhanced tropical upwelling. While rapid radiative adjustments from ozone and stratospheric water vapour make an essential contribution to the effective CH4 radiative forcing, the radiative impact of the respective slow feedbacks is rather moderate. In line with this, the climate sensitivity from CH4 changes in this chemistry–climate model set-up is not significantly different from the climate sensitivity in carbon-dioxide-driven simulations, provided that the CH4 effective radiative forcing includes the rapid adjustments from ozone and stratospheric water vapour changes.