What happens when the Student changes his Home Organization, or even studies in a different country? The ESI would change then in some cases?

The rule is that the European Student Identifier is issued by the higher education institution where students are initially enrolled for their studies. Hence, if a student decides to enrol in another institution for studies (without proceeding through a mobility arrangement) then there is a high likelihood that the student will also receive a new identifier. In the context of nationally issued student eIDs, the change only arises when changing country.