Abstract: Vibrational thermal properties of CuZn2InTe4, AgZn2InTe4, and Cu2CdSnTe4, derived from binary II-VI zinc-blendes, are reported based on first-principles calculations. While the chalcogenide atoms in these materials have the same lattice positions, the cation atom arrangements vary, resulting in different crystal symmetries and subsequent properties. The compositional differences have important effects on the vibrational thermal characteristics of the studied materials, which demonstrate that low-frequency optical phonons hybridize with acoustic phonons and lead to enhanced phonon-phonon scattering and low lattice thermal conductivities. The phonon density of states, mode Gruneisen parameters, and phonon scattering rates are also calculated, enabling deeper insight into the microscopic thermal conduction processes in these materials. Compositional variations drive differences among the three materials considered here; nonetheless, their structural similarities and generally low thermal conductivities (0.5-4 W/mK at room temperature) suggest that other similar II-VI zinc-blende derived materials will also exhibit similarly low values, as also corroborated by experimental data. This, combined with the versatility in designing a variety of motifs on the overall structure, makes quaternary chalcogenides interesting for thermal management and energy conversion applications that require low thermal conductivity.