Are flight schools required to verify citizenship of all students? Is the requirement to confirm citizenship and provide a logbook endorsement required for those who already hold a pilot certificate?

Does another instructor's prior endorsement that confirms citizenship was verified act as a substitution for subsequent instructors, or does each instructor need to check and make their own endorsement verifying citizenship no matter what? This is with the assumption that there are multiple freelance CFIs rather than a flight school employing many CFIs who take turns teaching the same student.

If no, would that not imply that each flight review with a new instructor would need a new citizenship verification or are currency checks exempt?
You are required to verify the identity of people undergoing initial Sport Pilot, Recreational Pilot, Private Pilot, Instrument, and Multi Engine training.  Any other kind of dual instruction does not require verification of US citizenship or TSA approval of resident aliens.  See 49 CFR 1552.3(h).

The CFI must inspect the original documents and verify they are authentic.  The CFI must endorse the CFI’s own logbook with an endorsement per AC 61-65 indicating they have done such and verify the student is a US Citizen.  These records must be kept for a minimum of 5 years.  If the CFI cannot validate the student as a citizen, be done, they must submit the student’s information and immigration documents to the TSA and with for their approval prior to beginning flight training.

https://www.aopa.org/advocacy/pilots/alien-flight-training-program/us-citizens-seeking-flight-training