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 $t\bar{t}$ and Ht production. For the $t\bar{t}$ production, one top quark decays via $t\rightarrow Hq$. The proton–proton collision data set analysed amounts to $({140}\,{\hbox {fb}^{-1}})$ at $(\sqrt{s}={13}\,\hbox {TeV})$. No significant excess beyond Standard Model expectations is observed and upper limits are set on the $t\rightarrow Hq$ branching ratios at 95 % confidence level, amounting to observed (expected) limits of $\mathcal {B}(t\rightarrow Hu)<2.8\,(3.0) \times 10^{-4}$ and $\mathcal {B}(t\rightarrow Hc)<3.3\,(3.8) \times 10^{-4}$. Combining this search with other searches for $tHq$ flavour-changing neutral-current interactions previously conducted by ATLAS, considering $H\rightarrow b\bar{b}$ and $H\rightarrow \gamma \gamma $ decays, as well as $H\rightarrow \tau ^{+}\tau ^{-}$ decays with one or two hadronically decaying $\tau $-leptons, yields observed (expected) upper limits on the branching ratios of $\mathcal {B}(t\rightarrow Hu)<2.6\,(1.8) \times 10^{-4}$ and $\mathcal {B}(t\rightarrow Hc)<3.4\,(2.3) \times 10^{-4}$.