A search for leptoquark pair production decaying into $te^- \bar{t}e^+$ or $t\mu ^- \bar{t}\mu ^+$ in final states with multiple leptons is presented. The search is based on a dataset of pp collisions at $\sqrt{s}=13~\text {TeV} $ recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb$^{-1}$. Four signal regions, with the requirement of at least three light leptons (electron or muon) and at least two jets out of which at least one jet is identified as coming from a b-hadron, are considered based on the number of leptons of a given flavour. The main background processes are estimated using dedicated control regions in a simultaneous fit with the signal regions to data. No excess above the Standard Model background prediction is observed and 95% confidence level limits on the production cross section times branching ratio are derived as a function of the leptoquark mass. Under the assumption of exclusive decays into $te^{-}$ ($t\mu ^{-}$), the corresponding lower limit on the scalar mixed-generation leptoquark mass $m_{\textrm{LQ}_{\textrm{mix}}^{\textrm{d}}}$ is at 1.58 (1.59) TeV and on the vector leptoquark mass $m_{{\tilde{U}}_1}$ at 1.67 (1.67) TeV in the minimal coupling scenario and at 1.95 (1.95) TeV in the Yang–Mills scenario.