A search is presented for a heavy resonance decaying into a Z boson and a Higgs (H) boson. The analysis is based on data from proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 138 fb$^{-1}$, recorded with the CMS experiment in the years 2016-2018. Resonance masses between 1.4 and 5 TeV are considered, resulting in large transverse momenta of the Z and H bosons. Final states that result from Z boson decays to pairs of electrons, muons, or neutrinos are considered. The H boson is reconstructed as a single large-radius jet, recoiling against the Z boson. Machine-learning flavour-tagging techniques are employed to identify decays of a Lorentz-boosted H boson into pairs of charm or bottom quarks, or into four quarks via the intermediate H $\to$ WW* and ZZ* decays. The analysis targets H boson decays that were not generally included in previous searches using the H $\to$ $\mathrm{b\bar{b}}$ channel. Compared with previous analyses, the sensitivity for high resonance masses is improved significantly in the channel where at most one b quark is tagged.