Using an $e^{+}{e}^{-}$ sample of $20.3\,\rm fb^{-1}$ collected at a center-of-mass energy $\sqrt{s}=$ 3.773 GeV with the BESIII detector, we report measurements of several four-body hadronic decays of $D$ mesons. All studied decay modes are dominated by Cabibbo-favoured amplitudes. The absolute branching fractions are determined to be $ℬ(D^0\to K_S^0{K}_S^0{K}_S^0{\pi }^0)=({7.8}_{-2.0}^{+2.3}\pm 1.5)\times {10}^{-5}$, $ℬ(D^0\to K_S^0{K}^{+}{K}^{-}{\pi }^0)=({18.4}_{-2.5}^{+2.6}\pm 2.4)\times {10}^{-5}$, $ℬ(D^0\to K_S^0{K}_S^0{K}^{-}{\pi }^{+})=({12.9}_{-1.6}^{+1.7}\pm 1.3)\times {10}^{-5}$, $ℬ(D^0\to K_S^0{K}_S^0{K}^{+}{\pi }^{-})=({5.7}_{-1.1}^{+1.2}\pm 1.0)\times {10}^{-5}$, $ℬ(D^0\to K^{+}{K}^{-}{K}^{-}{\pi }^{+})=({17.4}_{-1.7}^{+1.8}\pm 2.2)\times {10}^{-5}$, $ℬ(D^{+}\to K_S^0{K}_S^0{K}_S^0{\pi }^{+})=({6.0}_{-1.7}^{+2.1}\pm 1.6)\times {10}^{-5}$, and $ℬ(D^{+}\to K_S^0{K}^{+}{K}^{-}{\pi }^{+})=({13.8}_{-2.2}^{+2.4}\pm 2.5)\times {10}^{-5}$. Furthermore, significant $\varphi$ signals are found in the decay channels involving a $K^{+}{K}^{-}$ pair, and their fractions are determined as $ℬ(D^0\to \varphi K_S^0{\pi }^0)/ℬ(D^0\to K_S^0{K}^{+}{K}^{-}{\pi }^0)=({61}_{-14}^{+15}\pm 10)\%$, $ℬ(D^0\to \varphi K^{-}{\pi }^{+})/ℬ(D^0\to K^{+}{K}^{-}{K}^{-}{\pi }^{+})=({71}_{-10}^{+10}\pm 13)\%$, and $ℬ(D^{+}\to \varphi K_S^0{\pi }^{+})/ℬ(D^{+}\to K_S^0{K}^{+}{K}^{-}{\pi }^{+})=({59}_{-19}^{+22}\pm 9)\%$. The branching fractions of $D^0\to K_S^0{K}_S^0{K}_S^0{\pi }^0$, $D^0\to K_S^0{K}^{+}{K}^{-}{\pi }^0$, and $D^{+}\to K_S^0{K}_S^0{K}_S^0{\pi }^{+}$ are measured for the first time, and those of $D^0\to K_S^0{K}_S^0{K}^{-}{\pi }^{+}$, $D^0\to K_S^0{K}_S^0{K}^{+}{\pi }^{-}$, $D^0\to K^{+}{K}^{-}{K}^{-}{\pi }^{+}$, $D^0\to \varphi K^{-}{\pi }^{+}$, and $D^{+}\to K_S^0{K}^{+}{K}^{-}{\pi }^{+}$ are measured with improved precision. The first uncertainties are statistical and the second are systematic.