A criminal case can only be prosecuted by the country where the offense occurred.  If the person is never extradicted and remains in the U.S., he will eventually be released because the U.S. does not have jurisdiction to prosecute another country's charges.  They only have the authority to honor any agreements with that country to send defendants accused of crimes back to other country.Your question implies that you think charges are here in the U.S..... only the extradition is here in the U.S.  If a country makes a decision not to extradite, then the extradiction proceding ends.  The exception to this would be if the defendant were in the states illegally-- if so, the U.S. could make the decision to deport the defendant-- but not because of the criminal charges, but, instead, because they are in the U.S. illegally.