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 leptonic decays of the top quark along with Higgs boson decays into two W bosons, two Z bosons or a tau(+)tau(-) pair. It focuses on final states containing either two leptons (electrons or muons) of the same charge or three leptons. The considered processes are t (t) over bar and Ht production. For the t (t) over bar production, one top quark decays via t -> Hq. The proton-proton collision data set analysed amounts to (140 fb(-1)) at (root s = 13 TeV). No significant excess beyond Standard Model expectations is observed and upper limits are set on the t -> Hq branching ratios at 95% confidence level, amounting to observed (expected) limits of B(t -> Hu) < 2.8 (3.0) x 10(-4) and B(t -> Hc) < 3.3 (3.8) x 10(-4). Combining this search with other searches for t Hq flavour-changing neutral-current interactions previously conducted by ATLAS, considering H -> b (b) over bar and H -> gamma gamma decays, as well as H -> tau(+)tau(-) decays with one or two hadronically decaying tau-leptons, yields observed (expected) upper limits on the branching ratios of B(t -> Hu) < 2.6 (1.8) x 10(-4) and B(t -> Hc) < 3.4 (2.3) x 10(-4).