A search for chargino$-$neutralino pair production in three-lepton final states with missing transverse momentum is presented. The study is based on a dataset of $s=13$ TeV $pp$ collisions recorded with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. No significant excess relative to the Standard Model predictions is found in data. The results are interpreted in simplified models of supersymmetry, and statistically combined with results from a previous ATLAS search for compressed spectra in two-lepton final states. Various scenarios for the production and decay of charginos ($\chi ~1\pm$) and neutralinos ($\chi ~20$) are considered. For pure higgsino $\chi ~1\pm \chi ~20$ pair-production scenarios, exclusion limits at 95% confidence level are set on $\chi ~20$ masses up to 210 GeV. Limits are also set for pure wino $\chi ~1\pm \chi ~20$ pair production, on $\chi ~20$ masses up to 640 GeV for decays via on-shell $W$ and $Z$ bosons, up to 300 GeV for decays via off-shell $W$ and $Z$ bosons, and up to 190 GeV for decays via $W$ and Standard Model Higgs bosons.