can I use migration assistant to get old emails from time machine

Old emails will only transfer if they were "on my mac" at the time of the Time Machine backup.

IMAP, Exchange mailboxes that were listed at your email address are not stored on the Mac, thus do not get backed up to your Time Machine backup, and your email address company yourname@companyemailname may have backups they can recover and put back into your inbox.

Time Machine will recover stuff that gets stored as user data in your Users -> yourname folder, when you migrate that user to the new migration destination.

In Finder navigate to...

/Users/YourUserName/Library/Mail

Then Enter Time Machine.