The paper outlines results of a study of mixed free and forced convection in vertical flat and circular microchannels occupied with porous medium subject to slip boundary conditions. The problem was solved using analytical and numerical methodology including the lattice Boltzmann method. Effects of the Knudsen and Rayleigh numbers and porosity on velocity and temperature profiles, as well as the normalized Nusselt number and friction factor were elucidated. It was revealed that Knudsen number effects dominate in the vicinity of the wall, whereas the Rayleigh number effects are more significant near the channel axis. For high Rayleigh numbers, velocity profiles exhibit M-shapes having a point of minimum at the channel axis. In the same time, temperature profiles become uniform, so that the fluid temperature across the channel is almost equal to the wall temperature. An increase in porosity causes a decrease in the temperature and velocity jumps on the wall. It has been shown that for low Rayleigh numbers a decrease in the Darcy number causes an increase in heat transfer and hydraulic resistance. However, for high Rayleigh numbers, the trend becomes opposite. Comparisons of the quantitative results for the circular and flat channels revealed that the shape of channel cross section makes a significant influence on heat transfer and fluid flow. Results of the analytical studies were validated via comparisons with numerical simulations using the lattice Boltzmann method, which proved to be an accurate and powerful tool able to simulate mixed convection in microchannels, with the deviation from the analytical solution not exceeding 1%.