An analysis of the flavour oscillations of the charmed neutral meson is presented. The ratio of D⁰ → K⁺π⁻ and D⁰ → K⁻π⁺ decay rates is measured as a function of the decay time of the D⁰ meson and compared with the charge-conjugated system to search for charge-parity violation. The meson flavour at production is double-tagged by the charges of the muon and pion in the preceding B‾ → D* (2010)⁺π‾X and D^∗(2010)⁺ → D⁰π⁺ decays, respectively. These decays are selected from proton-proton collision data collected by the LHCb experiment at a centre-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 5.4 fb⁻¹. The flavour oscillation parameters, relating to the differences in mass and width of the mass eigenstates, are found to be y′ = (5.8 ± 1.6) × 10⁻³ and (x′)² = (0.0 ± 1.2) × 10⁻⁴. No evidence for charge-parity violation is seen either in the flavour oscillations or in the decay, where the direct charge-parity asymmetry is measured to be A_D = (2.3 ± 1.7) %.