The Constitutional implications of such an action are rather simple.
The English version of the Chilean Constitution (link) states in Chapter 4, Article 29 that the role of President will devolve on a designated Minister (currently the Minister of the Interior), as Vice President, in the event the office of President falls vacant. If the Vice President is not able to serve, the duty will pass to other ministers, then (individually) the Presidents of the Senate, Chamber of Deputies and Supreme Court.
If the original expiration of the President's term is less than 2 years away, the Chilean Congress, by absolute majority, will elect a new President within 10 days to serve out the term. The new President will take over from the Vice President within 30 days.
If the original expiration of the President's term is more than 2 years away, the Vice President must call for Presidential elections within their first 10 days in office, with elections happening on the first Sunday after the call is issued. 10 days after the election, the new President takes office.
Cabinet changes would be a matter of opinion, but Chapter 4, Article 32 of the Chilean Constitution does state that the President may appoint and dismiss ministers at will, so there would be no barrier to that occuring, if that was to be done by the Vice President or new President.