Impeachment and removal are an exclusive function of the Legislative branch. Neither the Judiciary nor the Executive can put limits on it. If Congress wants to, they can impeach and find guilty past officers as a punishment.
“If an official is impeached by the House and convicted by the requisite vote in the Senate, then Article I, Section 3, Clause 7, provides that the person convicted is further barred from any ‘Office of honor, Trust or Profit under the United States.’ The convicted official also loses any possible federal pensions.”
How would that be a punishment?
Impeachment is limited to removal from office (alas, no Charles I scenarios allowed ;)).
But removal is pointless if the target is already out of office.