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 τ+τ- pair. It focuses on final states containing either two leptons (electrons or muons) of the same charge or three leptons. The considered processes are tt¯ and Ht production. For the tt¯ production, one top quark decays via t→Hq. The proton–proton collision data set analysed amounts to (140fb-1) at (s=13TeV). 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)