  A measurement of $CP$-violating observables associated with the interference
of $B^0\to D^0 K^{*}(892)^0$ and $B^0\to \bar{D}^0 K^*(892)^0$ decay amplitudes
is performed in the $D^0 \to K^{\mp}\pi^{\pm}(\pi^+\pi^-),$ $D^0 \to
\pi^+\pi^-(\pi^+\pi^-)$, and $D^0\to K^+K^-$ final states using data collected
by the LHCb experiment corresponding to an integrated luminosity of $9$
$\text{fb}^{-1}$. $CP$-violating observables related to the interference of
$B^0_s\to D^0 \bar{K}^*(892)^0$ and $B_s^0\to \bar{D}^0 \bar{K}^*(892)^0$ are
also measured, but no evidence for interference is found. The $B^0$ observables
are used to constrain the parameter space of the CKM angle $\gamma$ and the
hadronic parameters $r_{B^0}^{DK^*}$ and $\delta_{B^0}^{DK^*}$ with inputs from
other measurements. In a combined analysis, these measurements allow for four
solutions in the parameter space, only one of which is consistent with the
world average.
