From Elvanör's Technical Wiki
- Follow this procedure:
- Create a label in Gmail with all the data you want to backup / remove (for instance: Gmail 2016).
- Label all the necessary messages with this label (the before:2016/01/01 filter may be useful).
- Launch Thunderbird, add your Gmail account via IMAP.
- Select the created label, select all messages inside it and right click -> Get selected messages (so that Thunderbird downloads the actual content, not just the header). This will take a very long time.
- Put Thunderbird in offline mode (important).
- Create a local folder (named for instance Gmail 2016), then choose all the selected messages inside the label, and Use "Copy the messages" to copy them to the local folder (this will take a very long time).
- Once this is done, you can also easily create a file backup (of several GBs) of this folder via the Import/Export tools (this is a Thunderbird plugin).
- At last, put all messages on the Gmail label to the Trash and empty the trash.
- Note that if you move the messages with Thunderbird (to a local folder), they won't be deleted on Gmail. By default Gmail archives them which just means the "Inbox" label is removed.