  We investigate the prospects for discovering the Flavour Changing Neutral
Current (FCNC) $tqZ$ couplings via two production processes yielding trilepton
signals: top quark pair production $pp\to t\bar{t}$ with one top decaying to
the $Z$ boson and one light jet and the anomalous single top plus $Z$ boson
production process $pp\to tZ$. We study these channels at various successors of
the Large Hadron Collider~(LHC), i.e., the approved High-Luminosity LHC
(HL-LHC) as well as the proposed High-Energy LHC~(HE-LHC) and Future Circular
Collider in hadron-hadron mode (FCC-hh). We perform a full simulation for the
signals and the relevant Standard Model (SM) backgrounds and obtain limits on
the Branching Ratios (BRs) of $t\to qZ~(q=u,c)$, eventually yielding a
trilepton final state through the decay modes $t\to b W^{+}\to
b\ell^{+}\nu_{\ell}$ and $Z\to \ell^{+}\ell^{-}$. The upper limits on these
FCNC BRs at 95\% Confidence Level (CL) are obtained at the HL-LHC with $\sqrt
s=14$ TeV and 3 ab$^{-1}$, at the HE-LHC with $\sqrt s=27$ TeV and 15 ab$^{-1}$
as well as at the FCC-hh with $\sqrt s=100$ TeV and 30 ab$^{-1}$.
