Here are instructions for exporting multiple emails using the most common email programs. 

• Apple Mail
• Gmail
• Microsoft Outlook

Apple Mail

1. Search for all messages matching your criteria.
2. Select all of them.
3. File > Save As...
4. Format > Raw Message Source
5. Create a new destination folder and click "Save."
6. If not still selected, re-select all the messages.
7. File > Save Attachments...

Note: in testing, exporting thousands of messages failed. Your mileage may vary.

Gmail​

1. In Gmail in a browser, find each email you want to export and select it or them (shift-click for contiguous or command-click for non-contiguous selections).
2. Right-click on the selection and choose Label as > create new
3. Create a label just for this purpose, like "export"
4. Search for, select and label any other messages with this new label.
5. When you've labeled all desired messages, open a new tab in your browser and go to https://takeout.google.com
6. In Google Takeout - below "Select data to include" click "Deselect all".
7. Scroll way down (or search in page) for Mail and select that.
8. In the mail section, click on All Mail data included and uncheck Include all messages in Mail.
9. Break for lunch. ;~)
10. In the Mail Content options list, click Select all and then Deselect all.
11. Scroll to the bottom of the list and check the label you just created and click OK.
12. Back in the takeout window scroll to the bottom of the page and click Next step.
13. In the Destination window, choose the default Send download link via email and Export once. Then Create export.
14. You will get an email with link when the file is ready to download.

Microsoft Outlook

1. Search for all messages matching your criteria.
2. Select the messages you wish to copy.  Shift+click selects a contiguous list of items. Ctrl+click selects multiple non-contiguous items.
3. Choose File > Save As > Outlook Message Format - Unicode.