A search for the pair-production of vector-like quarks optimized for decays into a Z boson and a third-generation Standard Model quark is presented, using the full Run 2 dataset corresponding to 139fb−1 of pp collisions at s=13TeV, collected in 2015–2018 with the ATLAS detector at the Large Hadron Collider. The targeted final state is characterized by the presence of a Z boson with high transverse momentum, reconstructed from a pair of same-flavour leptons with opposite-sign charges, as well as by the presence of b-tagged jets and high-transverse-momentum large-radius jets reconstructed from calibrated smaller-radius jets. Events with exactly two or at least three leptons are used, which are further categorized by the presence of boosted W, Z, and Higgs bosons and top quarks. The categorization is performed using a neural-network-based boosted object tagger to enhance the sensitivity to signal relative to the background. No significant excess above the background expectation is observed and exclusion limits at 95% confidence level are set on the masses of the vector-like partners T and B of the top and bottom quarks, respectively. The limits depend on the branching ratio configurations and, in the case of 100% branching ratio for T→Zt and 100% branching ratio for B→Zb, this search sets the most stringent limits to date, allowing mT>1.60TeV and mB>1.42TeV, respectively.