What is the process of water moving down through the soil called

The process is called Percolation.

Percolation (it means "filtering" or "passing through") commonly refers to the passage of a fluid through a more or less permeable medium.

The percolation of water through soils is studied in the context of urban ecology. For example to limit floods, and subsequently droughts.

when a reversible reaction is occurring in both directions at the same rate; the products and reactants do not change in concentration. a chemical reaction where the compound is broken into its constituent parts.