Kinetic and fluid dynamic processes in diode pumped alkali lasers (DPALs) are analyzed in detail using a semianalytical model, applicable to both static and flowing-gas devices. The model takes into account effects of temperature rise, excitation of neutral alkali atoms to high lying electronic states and their losses 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 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. It predicts the dependence of power on the flow velocity in flowing-gas DPALs and on the buffer gas composition. The maximum values of the laser power can be substantially increased by optimization of the flowing-gas DPAL parameters. In particular for the aforementioned 1 kW DPAL, 6 kW maximum power is achievable just by increasing the pump power and the temperature of the wall and the gas at the flow inlet (resulting in increase of the alkali saturated vapor density). Dependence of the lasing power on the pump power is non-monotonic: the power first increases, achieves its maximum and then decreases. The decrease of the lasing power with increasing pump power at large values of the latter is due to the rise of the aforementioned losses of the alkali atoms as a result of ionization. Work in progress applying two-dimensional computational fluid dynamics modeling of flowing-gas DPALs is also reported.