In this study, we introduced modifications to a prior existing enthalpic lattice Boltzmann method (LBM) tailored for simulating the conjugate heat transfer phenomena in non-homogeneous media with time-dependent thermal properties. Our approach is based upon the incorporation of the remaining terms of a conservative energy equation, excluding only the terms regarding flow compressibility and viscous dissipation, thereby accounting for the local and transient variations in the thermophysical properties. The solutions of verification tests, comprising assessments of both transient and steady-state solutions, validated the accuracy of the proposed model, further bolstering its reliability for analyzing heat transfer processes. The modified model was then used to perform an analysis on structured cavities under free convection, revealing compelling insights, particularly regarding transient regimes, demonstrating that the structured cavities exhibit a beneficial impact on enhancing the heat transfer processes, hence providing insights for potential design enhancements in heat exchangers. These results demonstrate the potential of our modified enthalpic LBM approach for simulating complex heat transfer phenomena in non-homogeneous media and structured geometries, offering valuable results for heat exchanger engineering and optimization.