Nothing gets out.
One switch holds every message the site tries to send. Turn it on before you import the database, not after.
You can still read what it tried to send.
Every held message is kept whole — recipients, subject, the rendered body, and the name of anything attached — so you can see exactly what would have gone out, and send it on if you decide it should.
It can’t be left off by accident.
Two constants in wp-config.php force the switch on and take it out of every administrator’s reach, including yours. While it is running, nothing inside WordPress can lift the hold — not the toolbar, not an administrator, not the REST API.