We present a comprehensive investigation of Partially-Saturated-Cells method (PSC) for the simulations of incompressible thermal flows. The methodology comprises an additional collision operator introduced into the conventional simplified thermal lattice-Boltzmann (STLBM) method that offers a unified evolution equation for solid and fluid media present in the computational domain. It is a type of diffuse-interface immersed-boundary method that is dictated by the volume fraction of the solid medium and facilitates a smooth transition of boundary conditions at the solid-fluid interface. In this work, we devise a mechanism to evaluate local heat flux at the solid-fluid interface and present its congruity with several heat transfer problems. We also derive the recovery of the macroscopic energy equation from PSC's evolution equation through Chapman-Enskog multiscale expansion. Our investigation also reveals that the method is first-order accurate for moving body simulations irrespective of the straight or curved boundary. It is also Galilean invariant and demonstrates identical heat transfer characteristics in stationary and moving reference frames. A comprehensive analysis of the PSC algorithm is presented to enlighten its suitability for predicting thermo-hydrodynamic characteristics in free, forced, and mixed convection phenomena involving both stationary and moving boundary scenarios. To the best of our knowledge, these aspects pertaining to PSC methodology have not been discussed in the literature so far. © 2023 Elsevier Ltd