Comprehensive analysis of kinetic and fluid dynamic processes in static and flowing-gas diode pumped alkali lasers (DPALs) using a semi-analytical and two- and-three dimensional computational fluid dynamics (2D and 3D CFD) models is reported. The models take into account effects of temperature rise and losses of alkali atoms due to ionization and chemical reactions, resulting in a decrease of the pump absorption, slope efficiency and lasing power. Effects of natural convection in static DPALs are also taken into account. The semi-analytical model is applied to Cs DPALs and the results are in good agreement with measurements in a static [B.V. Zhdanov, J. Sell and R.J. Knize, Electron. Lett. 44, 582 (2008)] and 1-kW flowing-gas [A.V. Bogachev et al., Quantum Electron. 42, 95 (2012)] DPALs. Dependence of the lasing power on the pump power is non-monotonic: the power first increases, achieves its maximum and then slowly decreases, the decrease being due to the rise of the losses of the alkali atoms as a result of ionization. Comparison of the semi-analytical and 2D CFD models shows that for the low pump power both models predict very close values of the laser power; however, at higher pump power, corresponding to saturation of the absorption of the pump transition, the values of the laser power calculated using the 2D CFD model are much higher than those obtained using the semianalytical model.