A search for pair-produced scalar and vector leptoquarks decaying into quarks and leptons of different generations is presented. It uses the full LHC Run 2 (2015–2018) data set of 139 fb−1 collected with the ATLAS detector in proton–proton collisions at a centre-of-mass energy of s = 13 TeV. Scalar leptoquarks with charge −(1/3)e as well as scalar and vector leptoquarks with charge +(2/3)e are considered. All possible decays of the pair-produced leptoquarks into quarks of the third generation (t, b) and charged or neutral leptons of the first or second generation (e, μ, ν) with exactly one electron or muon in the final state are investigated. No significant deviations from the Standard Model expectation are observed. Upper limits on the production cross-section are provided for eight models as a function of the leptoquark mass and the branching ratio of the leptoquark into the charged or neutral lepton. In addition, lower limits on the leptoquark masses are derived for all models across a range of branching ratios. Two of these models have the goal of providing an explanation for the recent B-anomalies. In both models, a vector leptoquark decays into charged and neutral leptons of the second generation with a similar branching fraction. Lower limits of 1980 GeV and 1710 GeV are set on the leptoquark mass for these two models. [Figure not available: see fulltext.]