When sediments are deposited, do they sink into the earth? I was listening to a geology podcast describing how sedimentary rock forms and later turns into metamorphic rock. As a river lays down sediment, the lower layers can eventually be several kilometers below ground, where heat and pressure are significant.

How does this happen? Does the river bed get built higher and higher until the first layer of sediment is buried deeply, or does the bottom layer get pushed down into the earth? In other words, is the elevation of a given section of river increasing while this happens, or does it remain the same?
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