The lift tube and air line create the driving force for the flow.
Water always tries to reach an equilibrium and maintain the same level. I am sure you probably played with a straw and a liquid refreshment. When you stopped drinking throught the straw, did you notice the fluid in side went back down to the same level as in the glass?
In an air driven UGF, the air bubbles displace water in the lift tube. This lowers the total weight of the water in the tube, as compared to the weight of the water for a similar volume outside the tube.
Water outside the tube will push down, through the gravel, under the plates and up the tube. If you slowly lower your tank level as you do a water change, when the water is reaching below the tube top, you will see this flow action.
You do not need the cartridges and air line if you are using a powerhead to pull water up the tube. The power heads will increase flow dramatically as compared to the air lift method. This increase the amount of oxygen righ water getting to the bacteria colony living in the substrate.
You can also set up some powerheads in a reverse manner, to push water down the tubes, under the plates and up through the gravel. This keeps the substrate cleaner and pushes the debri into the water column to be removed by other means of mechanical filtration.
Some HOB/HOT/canisters can also be modified to drive UGF plates.