In the present work, a comprehensive analysis is made to understand the effect of velocity boundary conditions on the flow and thermal behaviour during mixed convection flow in a nanofluid-saturated porous square cavity. Two different velocity boundary conditions based on the movement of horizontal walls of the cavity are considered. The vertical fixed walls are differentially heated and the horizontal lids are thermally insulated. We have adopted the two-phase thermal lattice Boltzmann model (TLBM) for nanofluid system and modified this model to simulate nanofluid-filled porous medium by incorporating the Brinkman–Forchheimer-extended Darcy model. The current results provide good concordance with the published results computed through conventional numerical techniques. The detailed study of the heat transfer rate, entropy generation is made for discretely varying Richardson numbers (Ri) from 0.1 to 10 and Darcy numbers (Da) from 10−4 to 10−2 while maintaining Grashof number (Gr) at 104 and volume fractions of Cu nanoparticle (ϕ) less than equal to 5%. It is observed from the results that the optimal flow condition in terms of energy efficiency depends on the values of Ri and Da. From the viewpoint of both 1st and 2nd laws of thermodynamics, the performance of nanofluid is not satisfactory compared to the base fluid for current configurations as the augmentation of entropy generation with ϕ is more prominent compared to heat transfer enhancement.