  The decays $D\to K^-\pi^+\pi^+\pi^-$ and $D \to K^-\pi^+\pi^0$ are studied in
a sample of quantum-correlated $D\bar{D}$ pairs produced through the process
$e^+e^- \to \psi(3770) \to D\bar{D}$, exploiting a data set collected by the
BESIII experiment that corresponds to an integrated luminosity of 2.93
fb$^{-1}$. Here $D$ indicates a quantum superposition of a $D^0$ and a
$\bar{D}^0$ meson. By reconstructing one neutral charm meson in a signal decay,
and the other in the same or a different final state, observables are measured
that contain information on the coherence factors and average strong-phase
differences of each of the signal modes. These parameters are critical inputs
in the measurement of the angle $\gamma$ of the Unitarity Triangle in $B^- \to
DK^-$ decays at the LHCb and Belle II experiments. The coherence factors are
determined to be $R_{K3\pi}=0.52^{+0.12}_{-0.10}$ and $R_{K\pi\pi^0}=0.78 \pm
0.04$, with values for the average strong-phase differences that are
$\delta_D^{K3\pi}=\left(167^{+31}_{-19}\right)^\circ$ and
$\delta_D^{K\pi\pi^0}=\left(196^{+14}_{-15}\right)^\circ$, where the
uncertainties include both statistical and systematic contributions. The
analysis is re-performed in four bins of the phase-space of the $D \to
K^-\pi^+\pi^+\pi^-$ to yield results that will allow for a more sensitive
measurement of $\gamma$ with this mode, to which the BESIII inputs will
contribute an uncertainty of around 6$^\circ$.
