U dont pump air in, you flow it through.
H3dxb , sorry to disillusion you, but the air is both "pumped" in and "flowed" through. Pumped in from the engine bleeds and flow controlled by the outflow valves. Outflow is controlled to maintain the required differential pressure such as to have the selected cabin altitude, and at the same time not exceed max permissible differential pressure. Safety valves are fitted to relieve the pressure in the event of a malfunction. Also, the cabin altitude will be determined by the level of pressurisation. Sea level can be maintained up until Max Differential is reached, however it is usual for the cabin altitude to increase at a rate of around 500'/min in the climb. Likewise if the cabin altitude is at, say 6000' in the cruise, it will gradually decrease at a lesser rate than the aircraft descends so that it is at destination airfield altitude for arrival. Ideally, a cabin altitude decrease of around 300'/min is desired to minimize passenger discomfort.