This research explores the effects of thermal and solute buoyancy forces on the unsteady boundary-layer (BL) stream through an upright porous flat plate with heat generation. The investigated fluid is viscid, impermeable, and electrically conductive. Thermo and solute buoyancy forces cause the mechanisms of thermal and material transfer. The nonlinear time-dependent partial differential equations (PDEs) about continuity, momentum, energy, and concentration are examined through suitable modifications. After mathematical modeling, an explicit finite difference method (EFDM) is used to solve a group of nonlinear dimensionless PDEs and appropriate boundary conditions. The detailing of the EFDM procedure due to the considered model is well-stated step by step. The validity of EFDM coding is conducted by reproducing similar previously available results. The mathematical formulation, stability, and convergence explored are also recognized. Searching for an appropriate uniform meshing and steady state condition is performed carefully. The existence of the pertinent parameters in this finite difference modeling is also checked. The thermo-physical consequences of inflowing physical values (buoyancy force and heat generation) on the velocity, temperature, and concentration distribution are scrutinized. The deviations in local and average skin friction coefficients, heat, and material transfer rates are also examined for physical curiosity in engineering research. The above-illustrated model equations are numerically simulated using LAHEY FORTRAN 95 v6.2. For graphical representation, Tecplot 365, and for data analysis, Microsoft Excel 2023 are used. The significant finding of this research reveals that the fluid velocity upsurges as the buoyancy intensifies, and the rising temperature generation enhances the thermal-material transport rates. The observed behavior is attributed to the thermal buoyancy force, which produces a pressure gradient. To compare the current modeling result, we correlate these findings to those of other research available in the literature.