  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 139 fb$^{-1}$ of $pp$ collisions
at $\sqrt{s}=13$ TeV, 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. In the singlet model, the limits allow $m_T >
1.27$ TeV and $m_B > 1.20$ TeV. In the doublet model, allowed masses are $m_T >
1.46$ TeV and $m_B >1.32$ TeV. In the case of 100% branching ratio for
$T\rightarrow Zt$ and 100% branching ratio for $B\rightarrow Zb$, the limits
allow $m_T > 1.60$ TeV and $m_B > 1.42$ TeV, respectively.
