When sediments are deposited, do they sink into the earth?
The rivers transport most of the sediments to the coastal or marine basins where they accumulate. An exception are lacustrine basins, but most of world rivers sediments are deposited on shallow or deep marine basins.

When the stack of sediments is large, listric faults develop, and the bottom layer get pushed down into the earth. This process is called subsidence.



Source: apg-pro.com

As new sediments accumulate, faults are activated and the basin becomes larger. Elevation of a given section of the basin remains approximately the same, as this is a slow process. 

In the case the bottom layer reaches great depth, burial metamorphism can develop:


  "Burial Metamorphism: occurs when sedimentary rocks that had undergone diagenesis are buried
  even deeper. Diagenesis grades into burial metamorphism, a relatively
  mild type of metamorphism resulting from the heat and pressure exerted
  by overlying sediments and sedimentary rocks. Although partial
  alteration of the mineralogy and texture may occur, bedding and other
  sedimentary structures are usually preserved".
  
  Source: uh.edu