In this study, the sensitivity of future lepton colliders to WIMP dark matter is evaluated assuming WIMP pair production accompanied by a photon from initial state radiation, through which the process can be identified. A full detector simulation for the International Large Detector (ILD) concept at the International Linear Collider (ILC) is performed for a \com energy of 500\,GeV. Energy scales of up to $3$\,TeV can be tested for different effective operators for WIMP masses almost up to half the \com energy. The sensitivity benefits from the polarised beams, which can reduce the main SM background from neutrino pair production substantially. In addition, systematic uncertainties are shown to be significantly reduced by combining data with several different polarisation configurations. In comparison to a previous study, the reconstruction of the forward detectors has been improved and the systematic uncertainties are fully treated. The results are also extrapolated to other \com energies, luminosities and beam polarisations. This allows to provide results for the full ILC programme, i.e.\ from 250\,GeV to 1\,TeV, as well as to give approximate results for other planned lepton colliders.

30 pages, 21 figures