Flint formation is complex, and the details still subject of research and debate.
The main factor is not a change in the relative proportion of siliceous and carbonate plankton, but the influence of chemical processes in the sediment after deposition. The flints precipitated where carbonate was dissolved. The dissolution occurred at the boundary between oxygenated water moving from the surface downwards and anoxic water rich in hydrogen sulphide from decomposing organic matter (the redox boundary). The reaction between oxygen and sulphide makes the water acidic at the boundary and this allows the carbonate to dissolve. Silica, which is in solution throughout the chalk, is in a state where it is precipitated from water with high carbonate concentrations, so flint forms at the redox boundary, typically between 5 and 10 metres below the sea bed.
The repetitive layering of bands seen in the cliffs may be influenced by cycles in the rate of sedimentation, with bands forming when sedimentation paused and the position of the redox boundary was stable. A secondary factor may be changes in the chalk matrix that influenced flow rates of water, and hence the redox boundary location. These changes are the result of changes in sediment supply, and bioturbation by burrowing animals at the sea bed.
The process is described in a paper by Clayton, 1986.