Hey there! Good question! And an important one. Time machine typically backs up your emails, but it also depends on a few factors, such as exclusions from Time Machine, (As Steve626 said), but also your available local storage, and your settings, etc).
I’ve had much success by using Time Machine to restore my mail. Sometimes the On My Mac section, (the local mailboxes), decide not to show up. So either they’re hidden, (You have to hover the cursor to the right side of the mailbox on the left column to see the word “hide” or “show”. Or they weren’t imported but are still on your new Mac.
Either way, another great way to save your emails is by exporting the mailboxes individually! Especially if your email account uses POP and not IMAP, (You can check with your email provider), usually though if you go to Mail > Accounts though, you’ll see either POP or IMAP next to the account, you can also look up what the settings might be here:
https://support.apple.com/mail-settings-lookup
Note that the settings here might not be the current settings it’s always best to check with your email provider for the current settings of your particular account.
Here’s how you can both import and export your mailboxes:
https://support.apple.com/guide/mail/import-or-export-mailboxes-mlhlp1030/mac
Just click on the particular mailbox, (Inbox, Sent, etc), select Mailbox > Export Mailbox), choose a location to save it.
Later, if you’re missing some mail, you can easily select File > Import Mailboxes, and select the .mbox file you have sported before. (If you exported from Apple Mail just choose Apple Mail).
If, after migrating data, your still missing the On My Mac mailboxes and didn’t export them, you should still be able to import them from the ~/Mail folder,
I usually would first go to the hidden user library to check it out:
Open Finder, click Go on the top menu, and you press and hold Option on the keyboard, click Library under Go, (Have to hold Option for it to show),
Open the Mail folder > V7 (If earlier versions V6, V5 etc.), Open each of those folders, (Except Mail Data), until you see the name of the mailbox you’re missing, (Such as: MyArchiveMail.mbox), they’re usually all in the same folder.
Highlight then copy all of them, then go to Finder > Downloads, and paste them there, (Or anywhere else, I just do this so they don’t start syncing to a cloud service).
Then back in Mail, you could select again File > Import Mailboxes, then choose those .mbox files that you moved. They’ll go under an Import folder, but you could typically move them anywhere else for organizational purposes.
And of course, you can do the same import process and import from whatever location you saved those mailboxes to!
Note, that those same mailboxes should also be saved in Time Machine. If you navigate to that same location; ~/Mail/V7, then “Enter Time Machine, just click on the name of the mailbox and hit “Restore”, (This will replace any existing mailbox in the same location with the same name).
Anyhow, hope that helps! Just saying there is more than one way to backup your important mailboxes! I choose to have multiple backups for my important information, because I’ve been there, where I’ve lost very important data, and it’s not fun! Cheers!
https://support.apple.com/en-afri/guide/mac-help/mh11422/mac
