Only if you set the archive folder in Thunderbird to Gmails "all mail" folder. That is the canonical source in Googles world view and contains all mail which has no label. Personally I am something of a fan of Google takeout as a periodic backup and/or Archive of old Gmail. It allows for the Gmail accont to be treated as a current mail source and everything prior to the takeout download can be deleted from the GMail account. Just be sure to change the delete setting to not move mail into the deleted folder in GMail as doing so, will restore mail deleted from the all mail folder back there as a new copy is uploaded from your local deleted folder. Note that if you get takeout ZIp files by label, they can be imported to Thunderbird local folders (local machine storage) using the import export tools addon after they are unzipped to MBOX files. If you do not, the file is huge and the import is less structured. Still possible, but less fun. https://addons.thunderbird.net/en-US/thunderbird/addon/importexporttools-ng/?src=ss