A search for new heavy scalars with flavour-violating decays in final states with multiple leptons and b-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$_{tc}$, and ρ$_{tu}$). 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} $ = 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 boson m$_{H}$ between 200 − 630 GeV with couplings ρ$_{tt}$ = 0.4, ρ$_{tc}$ = 0.2, and ρ$_{tu}$ = 0.2 are excluded at 95% confidence level. Additional interpretations are provided in models of R-parity violating supersymmetry, motivated by the recent flavour and (g − 2)$_{μ}$ anomalies.