In the United States, when a new President takes office he or she has the power to appoint a new Chief of Justice that will serve during that term until the President leaves the office, whatever reason it may be: resignation, death, conviction, etc.
Such power is guaranteed in Section 2, Article II of the Constitution. And even though it is the president who has the power to appoint the Chief of justice, it has to be approved by the Senate.
So, what happens to the Chief of Justice is that he or she is replaced for a new Chief of Justice, that the new President shall appoint.