A search is presented for flavour-changing neutral-current interactions involving the top quark, the Higgs boson and an up-type quark (q = u, c) with the ATLAS detector at the Large Hadron Collider. The analysis considers lep-tonic decays of the top quark along with Higgs boson decays into two W bosons, two Z bosons or a τ +τ − pair. It focuseson final states containing either two leptons (electrons or muons) of the same charge or three leptons. The considered processes are t ¯t and H t production. For the t ¯t production, one top quark decays via t → H q. The proton–proton collision data set analysed amounts to (140 fb−1) at (√s = 13 TeV). No significant excess beyond Standard Model expectations is observed and upper limits are set on the t → H q branch-ing ratios at 95 % confidence level, amounting to observed (expected) limits of B(t → H u) < 2.8 (3.0) × 10−4 and B(t → H c) < 3.3 (3.8) × 10−4 . Combining this search with other searches for t H q flavour-changing neutral-current interactions previously conducted by ATLAS, consideringH → b ¯b and H → γ γ decays, as well as H → τ +τ −decays with one or two hadronically decaying τ -leptons, yields observed (expected) upper limits on the branch ingratios of B(t → H u) < 2.6 (1.8) × 10−4 and B(t → H c) <3.4 (2.3) × 10−4 .