Can I import exchange pst files into exchange mailboxes using impersonation credentials?

We are migrating from a hosted exchange solution back to an on premise Exchange 2013.

The tools we have tried to use are not working well. So we have decided to go at it manually. We are planning to export the mailboxes from the hosted solution to PST files.

To import them we need to make sure that we able to pair them up to the users' mailboxes in exchange. Are we able to do this without requiring the users' credentials? Perhaps using impersonation or setting administrator privileges on each mailbox for an admin account?

Thanks in advance.