Would the ESI change if the Student changes his Home Organization, or even studies in a different country

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.