The five appear as a bare list in the Great Plan chapter of the Book of Documents, with no cycles and no correspondences attached. What turned that list into a system was the work of the Warring States and Han periods, when thinkers associated with the school of yin, yang and the five phases, Zou Yan chief among them by later report, matched the five to directions, seasons, colours, tastes, planets, organs, numbers and the succession of dynasties. Under the Han, Dong Zhongshu built the resulting scheme into a political theology in which the conduct of the ruler and the condition of the natural world answered to one another.
Joseph Needham called this way of reasoning correlative thinking, and his second volume on the history of scientific thought in China remains the fullest account of it in English. Its logic is not causal in the sense a modern laboratory would recognise. Things in the same category resonate, and a change in one register shows up in another because they belong together, not because one pushed the other.
The dynastic use is the sharpest example. A ruling house claimed a phase, and the phase that overcame or generated it determined which house could legitimately follow. Cosmology and succession were one argument. That is why the scheme survived as the organising frame of Chinese medicine, divination, music theory, statecraft and the calendar rather than as one theory among several: it was the shared grammar in which claims in all of those fields were made.