  Based on 2.93~fb$^{-1}$ $e^+e^-$ collision data taken at center-of-mass
energy of 3.773 GeV by the BESIII detector, we report the measurements of the
absolute branching fractions of $D^0\to K^+K^-\pi^0\pi^0$, $D^0\to
K^0_SK^0_S\pi^+\pi^-$, $D^0\to K^0_SK^-\pi^+\pi^0$, $D^0\to
K^0_SK^+\pi^-\pi^0$, $D^+\to K^+K^-\pi^+\pi^0$, $D^+\to K^0_SK^+\pi^0\pi^0$,
$D^+\to K^0_SK^-\pi^+\pi^+$, $D^+\to K^0_SK^+\pi^+\pi^-$, and $D^+\to
K^0_SK^0_S\pi^+\pi^0$. The decays $D^0\to K^+K^-\pi^0\pi^0$, $D^0\to
K^0_SK^-\pi^+\pi^0$, $D^0\to K^0_SK^+\pi^-\pi^0$, $D^+\to
K^0_SK^0_S\pi^+\pi^0$, and $D^+\to K^0_SK^+\pi^0\pi^0$ are observed for the
first time. The branching fractions of the decays $D^0\to
K^0_SK^0_S\pi^+\pi^-$, $D^+\to K^+K^-\pi^+\pi^0$, $D^+\to K^0_SK^-\pi^+\pi^+$,
and $D^+\to K^0_SK^+\pi^+\pi^-$ are measured with improved precision compared
to the world-average values.
