If you are talking about an imap mail account: The folders show a virtual view of what is on server. You would need to get a copy of emails off the server first.

synchronise folders for download and offline use. https://support.mozilla.org/en-US/kb/imap-synchronization

then in Offline mode use: Right click on highlighted emails and select 'Copy to' and select a suitable folder in 'Local Folders' mail account. If you can read copied emails in 'Local Folders' mail account then you have a good copy. Delete the copied emails from the imap mail account folder. When you go back into online mode, folders will synchronise with server and delete them off server.

This is too complicated for ordinary people. I followed the sync instructions and, when offline, moved files to a local folder. They moved, and now they're gone.

The simple answer is if you have a single devoce to get mail on and have no desire for server based copies you would be better server with a POP account. Please see https://support.mozilla.org/en-US/kb/glossary-terms-including-types-accounts#w_mail-accounts

Neither IMAP or POP is superior and you need a basic knowledge of both to determine which provides the features that will best serve you. I use a mixture of IMAP and POP mail accounts, some I delete mail from the server on download using pop, others I use IMAP on as I also read them using my phone.