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