In semiconductors almost all heat is conducted by phonons (lattice vibrations), which is limited by their quasi-particle lifetimes. Phonon-phonon interactions represent scattering mechanisms that produce thermal resistance. In thermoelectric materials, this resistance due to anharmonicity should be maximised for optimal performance. We use a first-principles lattice-dynamics approach to explore the changes in lattice dynamics across an isostructural series where the average atomic mass is conserved: ZnS to CuGaS2 to Cu2ZnGeS4. Our results demonstrate an enhancement of phonon interactions in the multernary materials and confirm that lattice thermal conductivity can be controlled independently of the average mass and local coordination environments.