  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 TeV recorded
with the ATLAS detector during Run 2 of the Large Hadron Collider,
corresponding to an integrated luminosity of 139~fb$^{-1}$. Events are selected
with two or more 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. Four signal
regions, with the requirement of at least three light leptons, 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
$m_{\mathrm{LQ}_{\mathrm{mix}}^{\mathrm{d}}}$ 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.
