  We explore the experimental sensitivities of measuring the $gg \rightarrow Z
\gamma$ process at the LHC to the dimension-8 quartic couplings of gluon pairs
to the $Z$ boson and photon, in addition to comparing them with the analogous
sensitivities in the $gg \to \gamma \gamma$ process. These processes can both
receive contributions from 4 different CP-conserving dimension-8 operators with
distinct Lorentz structures that contain a pair of gluon field strengths, $\hat
G^a_{\mu \nu}$, and a pair of electroweak SU(2) gauge field strengths,
$W^i_{\mu \nu}$, as well as 4 similar operators containing a pair of $\hat
G^a_{\mu \nu}$ and a pair of U(1) gauge field strengths, $B_{\mu \nu}$. We
calculate the scattering angular distributions for $gg \rightarrow Z \gamma$
and the $Z \to \bar f f$ decay angular distributions for these 4 Lorentz
structures, as well as the Standard Model background. We analyze the
sensitivity of ATLAS measurements of the $Z(\to \ell^+\ell^-, \bar \nu \nu,
\bar q q)\gamma$ final states with integrated luminosities up to 139 fb$^{-1}$
at $\sqrt{s} = 13$ TeV, showing that they exclude values $\lesssim 2$ TeV for
the dimension-8 operator scales, and compare the $Z \gamma$ sensitivity with
that of an ATLAS measurement of the $\gamma \gamma$ final state. We present
combined $Z \gamma$ and $\gamma \gamma$ constraints on the scales of
dimension-8 SMEFT operators and $\gamma \gamma$ constraints on the nonlinearity
scale of the Born-Infeld extension of the Standard Model. We also estimate the
sensitivities to dimension-8 operators of experiments at possible future
proton-proton colliders with centre-of-mass energies of 25, 50 and 100 TeV, and
discuss possible measurements of the $Z$ spin and angular correlations.
