Abstract Measurements of CP observables in B0 → DK∗0 decays are presented, where D represents a superposition of D0 and D¯0$$ {\overline{D}}^0 $$ states. The D meson is reconstructed in the two-body final states K+π−, π+K−, K+K− and π+π−, and, for the first time, in the fourbody final states K+π−π+π−, π+K−π+π− and π+π−π+π−. The analysis uses a sample of neutral B mesons produced in proton-proton collisions, corresponding to an integrated luminosity of 1.0, 2.0 and 1.8 fb−1 collected with the LHCb detector at centre-of-mass energies of s=7,8$$ \sqrt{s}=7,8 $$ and 13 TeV, respectively. First observations of the decays B0 → D(π+K−)K∗0 and B0 → D(π+π−π+π−)K∗0 are obtained. The measured observables are interpreted in terms of the CP -violating weak phase γ.