A search is presented for new physics in events with two low-momentum, oppositely charged leptons (electrons or muons) and missing transverse momentum in proton-proton collisions at a centre-of-mass energy of 13 TeV. The data collected using the CMS detector at the LHC correspond to an integrated luminosity of 35.9 fb$^{-1}$. The observed event yields are consistent with the expectations from the standard model. The results are interpreted in terms of pair production of charginos and neutralinos ($\widetilde{\chi}^\pm_1$ and $\widetilde{\chi}^0_2$) with nearly degenerate masses, as expected in natural supersymmetry models with light higgsinos, as well as in terms of the pair production of top squarks ($\widetilde{\mathrm{t}}$), when the lightest neutralino and the top squark have similar masses. At 95% confidence level, wino-like $\widetilde{\chi}^\pm_1$/$\widetilde{\chi}^0_2$ masses are excluded up to 230 GeV for a mass difference of 20 GeV relative to the lightest neutralino. In the higgsino-like model, masses are excluded up to 168 GeV for the same mass difference. For $\widetilde{\mathrm{t}}$ pair production, top squark masses up to 450 GeV are excluded for a mass difference of 40 GeV relative to the lightest neutralino.