The rigorous treatment of four-particle intermediate and final states poses a major challenge for lattice calculations of scattering and decay amplitudes, as well as long-distance matrix elements. As a step towards addressing these challenges, we present a new formalism that perturbatively relates two- and four-particle finite-volume energies and matrix elements to the couplings of the infinite-volume theory. Our method works at leading order in the two-to-two, four-to-four, and two-to-four couplings of the theory, while also capturing the leading finite-volume effects associated with two-to-two subprocess scattering in the four-particle sector. The result takes the form of a quantization condition which we implement numerically to produce a plot of volume-dependent energies for center-of-mass energies up to the six-particle threshold. The solutions exhibit a clear signature of two- and four-particle-like states and the avoided level crossings between them, which are particularly sensitive to the two-to-four coupling. We further discuss the implications of this formalism for quantifying the four-particle contributions in decay and transition amplitudes (e.g. for hadronic $D$ decays).