You mention 'Gmail'. Are you talking about a gmail POP or IMAP mail account ?

I'm going to assume it is IMAP, so please correct me if it is POP.

IMAP accounts store all emails on the server. You subscribe to see the server Folders. Imap folders synchronise with server folders of same to display whatever is on the server. You can choose to download full copies of emails or just headers and that can be done on a per folder basis. As imap account folders can only display what is on server and they are synchronising with server, you cannot regard an imap folder as an independent copy. But you can use it as a good enough copy in order to create an independent backup of emails.

Gmail stores all emails, incoming, sent - everything in the 'All Mail' folder. Gmail knows that this is not much use and very difficult to work with, so it applies 'labels' (like a special tag) to those emails which means it can create a more traditional view of virtual folders which people understand eg: Inbox, Sent, Spam etc.

If you only remove a 'label' the email is not deleted, it is still in 'All Mail' folder. Gmail call this 'Archiving'. So, if you select to Archive then all labels are removed and email cannot be seen in anything other than 'All Mail'.

Most people do not bother to subscribe to see the 'All Mail' folder because it is massive and is only a complete duplicate of everything. It's also a pain trying to work out what emails are actually archived, because there is no difference between the archived mail or any other email.

One way around this, is to create a new folder in the gmail imap account and call it 'Archives'. Gmail will treat this as a new label. So in webmail account you would see a new folder/label called Archives.

In Account Settings > Copies & Folders for the gmail imap account: Select 'Keep message archives in:' Select 'Other' and choose the 'Archives on gmail email address'

The emails are easier to sort and locate. The emails are still stored on the gmail server and so are still visible via webmail.

The 'Local Folders' account is a special account in Thunderbird. You can use it to store any emails. The emails in this account are stored on your computer and are completely independent of any server. If you try to 'Move' an email from an imap mail account folder into a folder - let's call it 'Old' in 'Local Folders', there is communication between Thunderbird and the server. As the email is moved out of the imap folder, but not deleted, it means the 'label' is removed and therefore email is now only visible in 'All Mail'. However, if the move went ok then you will also see it in 'Local Folders' account in the 'Old' folder. This will be a completely independent copy, but make sure you have 'moved' a fully downloaded copy and not just a header.

Personally, I would always advise you use the right click > 'Copy to' option because if something goes wrong then you still have a good copy on server.

Later if you do not want to keep the copy on server, you can use 'Delete' and this puts the email into the imap Trash folder. Gmail will auto delete any emails that have been put into the gmail Trash after 30 days. That means it is deleted from the 'All Mail' folder.

IF you want to keep a copy in eg: 'Archives' in 'Local Folders' mail account:

Create a folder called 'Archives' in the 'Local Folders' mail account. It is not created by default.

Check your gmail account is set up to download full copies for all folders no matter the age of email. In 'Account Settings' > 'Synchronisation & Storage' for imap gmail account Message Synchronising

• select checkbox 'keep messages in all folders for this account on this computer'
• click on 'Advanced...' button
• Make sure all folders and subfolders are checkboxed for download - click on Ok

Under 'Disk Space'

• Select 'Synchronise all messages locally regardless of age'

In 'Account Settings' > 'Copies & Folders' for the gmail imap account:

• Select 'Keep message archives in:'
• Select 'Other' and choose the 'Archives on Local Folders'

This will put a copy into the 'Archives' folder in the 'Local Folders' mail account. It will remove the email from whatever imap folder it was in. It will remove all labels from that email and gmail will archive it in the 'All Mail' folder.

Please remember to frequently compact folders to clean up old hidden traces of moved and deleted emails.

Thanks to those who offered their assistance. As it turns out, I had two separate issues. The first (coincidence?) occurred close to the last Windows 10 upgrade. For whatever reason, my "Local" folder got wiped out ; that was where I kept my archived folders.

I figured out that Google must have then simply kept the recently archived emails in the "All" folder because it removed the label they had once I put them in the T'bird archives.

Bottom line is that I was able to get things back to normal while I waited for responses. But your kindness is appreciated.