Alloys based on lanthanum phosphate (LaPO4) are often employed as thermal barrier coatings, due to their low thermal conductivity and structural stability over a wide temperature range. To enhance the thermal-insulation performance of these alloys, it is essential to comprehensively understand the fundamental physics governing their heat conduction. Here, we employ the Wigner formulation of thermal transport in conjunction with first-principles calculations to elucidate how the interplay between anharmonicity and compositional disorder determines the thermal properties of La1-xGdxPO4 alloys, and discuss the fundamental physics underlying the emergence and coexistence of particle-like and wavelike heat-transport mechanisms. We also show how the Wigner transport equation correctly describes the thermodynamic limit of a compositionally disordered crystal, while the Boltzmann transport equation does not. Our predictions for microscopic vibrational properties (temperature-dependent Raman spectrum) and for macroscopic thermal conductivity are validated against experiments. Finally, we leverage these findings to devise strategies to optimize the performance of thermal barrier coatings.