Precision measurements of diboson production at the LHC is an important probe of deviations from the Standard Model. The gluon-fusion channel of this process offers a connection between the Higgs and top sectors. We study in a systematic way gluon-induced diboson production in the Standard Model Effective Field Theory, with a focus on double $Z$ and associated $ZH$ production. We compute the helicity amplitudes of these processes at one loop and with up to one insertion of a dimension-6 operator and we study their high-energy limit in order to identify which operators in each channel lead to growths with energy and for which helicity configurations. In addition we perform a phenomenological study of $ZH$ production, including both quark and gluon initial states. We show that for some top operators the gluon-induced channel can offer competitive sensitivity to constraints obtained from top quark production processes.