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Leptonic signatures of Dark Matter (DM) are one of the cleanest ways to discover such a secluded form of matter at high energy colliders. We explore the full parameter space relevant to multi-lepton (2- and 3-lepton) signatures at the Large Hadron Collider (LHC) from representative minimal consistent models with scalar and fermion DM. In our analysis, we suggest a new parametrisation of the model parameter spaces in terms of the DM mass and mass differences between DM and its multiplet partners. This parametrisation allows us to explore properties of DM models in their whole parameter space. This approach is generic and quite model-independent since the mass differences are related to the couplings of the DM to the Standard Model (SM) sector. We establish the most up-to-date LHC limits on the inert 2-Higgs Doublet Model (i2HDM) and Minimal Fermion DM (MFDM) model parameter spaces, by using the complementary information stemming from 2- and 3-lepton signatures. We provide a map of LHC efficiencies and cross-section limits for such 2- and 3-lepton signatures allowing one to easily make model-independent reinterpretations of LHC results for analogous classes of models. We also present combined constraints from the LHC, DM relic density and direct search experiments indicating the current status of the i2HDM and MFDM model.