A search for new heavy scalars with flavour-violating decays in final states with multiple leptons andb-tagged jets is presented. The results are interpreted in terms of a general two-Higgs-doublet model involving an additional scalar with couplings to the top-quark and the three up-type quarks (ρ_tt,ρ_tc, andρ_tu). The targeted signals lead to final states with either a same-sign top-quark pair, three top-quarks, or four top-quarks. The search is based on a data sample of proton-proton collisions at$$ \sqrt{s} $$$\sqrt{s}$= 13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb⁻¹. Events are categorised depending on the multiplicity of light charged leptons (electrons or muons), total lepton charge, and a deep-neural-network output to enhance the purity of each of the signals. Masses of an additional scalar bosonm_Hbetween 200 − 630 GeV with couplingsρ_tt= 0.4,ρ_tc= 0.2, andρ_tu= 0.2 are excluded at 95% confidence level. Additional interpretations are provided in models ofR-parity violating supersymmetry, motivated by the recent flavour and (g −2)_μanomalies.