Charm meson oscillations are observed in a time-dependent analysis of the ratio of ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ to ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ decay rates, using data corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ recorded by the LHCb experiment. The measurements presented are sensitive to the phase-space averaged ratio of doubly Cabibbo-suppressed to Cabibbo-favored amplitudes ${r}_{D}^{K3\ensuremath{\pi}}$ and the product of the coherence factor ${R}_{D}^{K3\ensuremath{\pi}}$ and a charm mixing parameter ${y}_{K3\ensuremath{\pi}}^{\ensuremath{'}}$. The constraints measured are ${r}_{D}^{K3\ensuremath{\pi}}=(5.67\ifmmode\pm\else\textpm\fi{}0.12)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, which is the most precise determination to date, and ${R}_{D}^{K3\ensuremath{\pi}}{y}_{K3\ensuremath{\pi}}^{\ensuremath{'}}=(0.3\ifmmode\pm\else\textpm\fi{}1.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which provides useful input for determinations of the CP-violating phase $\ensuremath{\gamma}$ in ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}D{K}^{\ifmmode\pm\else\textpm\fi{}}$, $D\ensuremath{\rightarrow}{K}^{\ensuremath{\mp}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ decays. The analysis also gives the most precise measurement of the ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ branching fraction, and the first observation of ${D}^{0}$--${\overline{D}}^{0}$ oscillations in this decay mode, with a significance of 8.2 standard deviations.