Afaik Italy can’t close its borders and deport back migrants because it’s not sovereign anymore and in the eu
That's what Frontex is all about. Same in Lithuania - not only it can close outer borders, but it has a requirement to do so.
So unless being sovereign is about having borders open, this argument doesn't work.
During the 2015 migrant crisis, did not refugees end up traveling to Germany while first entering Greece and Italy? If Dublin regulation really applied, Germany would not need to take the responsibility of the majority of the asylum seekers. At the same time, that only concerns for the asylum seekers and not the unregistered aliens.
They can't do so permanently, but they can provide some sort of a justification every six months and keep it closed, nobody can veto them.