Theoretical influence of the buoyancy and thermal radiation effects on the MHD (magnetohydrodynamics) flow across a stretchable porous sheet were analyzed in the present study. The Darcy–Forchheimer model and laminar flow were considered for the flow problem that was investigated. The flow was taken to incorporate a temperature-dependent heat source or sink. The study also incorporated the influences of Brownian motion and thermophoresis. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. A favorable comparison with earlier published studies was achieved. Graphs were used to investigate and explain how different physical parameters affect the velocity, the temperature, and the concentration field. Additionally, tables are included in order to discuss the outcomes of the Sherwood number, the Nusselt number, and skin friction. The fundamental governing partial differential equations (PDEs), which are used in the modeling and analysis of the MHD flow problem, were transformed into a collection of ordinary differential equations (ODEs) by utilizing the similarity transformation. A semi-analytical approach homotopy analysis method (HAM) was applied for approximating the solutions of the modeled equations. The model finds several important applications, such as steel rolling, nuclear explosions, cooling of transmission lines, heating of the room by the use of a radiator, cooling the reactor core in nuclear power plants, design of fins, solar power technology, combustion chambers, astrophysical flow, electric transformers, and rectifiers. Among the various outcomes of the study, it was discovered that skin friction surges for 0.3 ≤F1≤ 0.6, 0.1 ≤k1≤ 0.4 and 0.3 ≤M≤ 1.0, snf declines for 1.0 ≤Gr≤ 4.0. Moreover, the Nusselt number augments for 0.5 ≤R≤ 1.5, 0.2 ≤Nt≤ 0.8 and 0.3 ≤Nb≤ 0.9, and declines for 2.5 ≤Pr≤ 5.5. The Sherwood number increases for 0.2 ≤Nt≤ 0.8 and 0.3 ≤Sc≤ 0.9, and decreases for 0.1 ≤Nb≤ 0.7.