You can do a one-time transfer of all Google Mail messages or just those in certain label(s) using Google Takeout, which downloads your messages as an MBOX file. You can then use Thunderbird (or Apple Mail on a Mac) to access messages and attachments in the MBOX file and transfer them to another account, if you wish.

If you are transfering messages to someone else, for example the successor of a project, send the recipient the MBOX file so he or she can access it (If the MBOX file is larger than 25MB, you cannot email it, but you can share it through Google Drive).

I. Download Message(s) with Google Takeout

1. If you only want to download select message(s), start in your Google Mail by applying a label, for example "messages to download," to only the message(s) you want to download
2. Go to https://www.google.com/settings/takeout 
3. Click "Select None" (Thunderbird can only store your emails, it cannot store this other data)
4. Scroll down to "Mail", click on the gray X to the right
   1. If you want to download only certain messages, click "All Mail"
   2. Check "Select Labels"
   3. Check the labels that tag the emails you want to download
5. Click "Next"
6. Do not change the file type, click "Create Archive"
7. The zip will be sent via your selected delievery method (by default, you will get an email with a link to download the zip) - it may not be instant, the more emails you are downloading, the longer it will take to create your archive

For more information on creating achives, click here. 

II. Setup up Thunderbird with your Stony Brook Mail only

III. Install the Import/Export Extension in Thunderbird

This add-on will help you easily import the MBOX file and convert it into a folder housing your email messages and attachments

1. Download the ImportExportTools NG add-on from the Mozilla website: https://addons.thunderbird.net/en-US/thunderbird/addon/importexporttools-ng/
   *Ensure that this addon is compatible with the version of Thunderbird that you are using. Other versions of the add-on can be downloaded here
2. Go to  > Add-ons.  This will show you a list of all your Thunderbird addons.  If you just installed Thunderbird, it might be an empty list. 
3. Click Add-ons on this list to open the Add-on Manager page.
4. Click Browse all Add-ons.
5. Browse to the .xpi file (ImportExportTools NG add-on) you downloaded in step one and click Open. Give it a few moments to process the file, and when it's finished you can click on the Install Now button
6. Exit Thunderbird
7. Start Thunderbird again
8. Click on Local Folders to deposit the MBOX file there
9. Go to  > Tools > ImportExportTools > Import mbox file
10. With "Import directly one or more mbox files" selected, click OK
    
11. Navigate to the folder of extracted files that you'd like to move into Thunderbird; click Open
12. Your MBOX file will instantly appear on the left side pane of your screen (You may set up another email account in Thunderbird to drag the MBOX folder to that account)
    
13. To transfer the file to another account, repeat steps II.3- II.5 to add the other account, and then drag/drop the MBOX file to the newly added account