Transferring Contents Of Local Mailboxes In Mac Mail To Server-labeled Mailboxes

In a Q&A from a couple weeks ago you recommended against having local mailboxes in Mac Mail. I have a bunch of old local mailboxes On My Mac and intend on changing them over to server-based (right terminology?) gmail labels. My question concerns the contents of my existing local mailboxes. In many cases there are emails I created from several different email addresses that were collected in the subject matter local mailboxes. In some cases the original email addresses are now obsolete/gone. Can I move those old emails under a gmail label even if they were not originated in gmail? Will I be able to conduct searches for the old non-gmail emails?
Thanks, Julie
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Julie S