You do not mention what sort of mail account you use in Thunderbird. Pop or IMAP? Are the emails on the server? check by accessing via webmail using a browser.

How did you access the emails from early May 2014 to late August 2014 ?

Pop account is what I use on both computers. I accessed the May to August email on my home computer. When I arrived at my second home I downloaded email and there is a block of missing email from May to August.

re:Pop account is what I use on both computers.

Pop mail accounts access the server Inbox and if it locates new mail not previously downloaded, it will download those emails to the Thunderbird mail account Inbox and by default it will delete the copy off the server.

You can set the Pop mail account to 'leave messages on the server'.

It sounds like your other computer used pop, but it was not configured to leave a copy on the server. This would mean tht as that computer downloaded emails, it was also deleteing the copy off the server. So when you used the second home computer, it cannot download something that does not exist.

There are two ways off checking this. Firstly, if you logon to your webmail account via a browser you will see if the emails are actually on the server.

Secondly, check both computers to see if the option to leave messages on the server is selected. I would suggest that you select this option so that you can download to both computers when using either.

This is where you make the 'leave messages on the server' setting. Tools > Account Settings > Server Settings for the mail account select 'leave messages on server'

if desired you can also select: 'for at most xx days' to remove older mail and / or 'Until I delete them' to remove deleted emails. click on OK to save changes.

To get emails saved on one computer pop mail account onto another computer pop mail account where the pop mail account is identical.

Download and install this addon on both computers.

• https://addons.mozilla.org/en-US/thunderbird/addon/importexporttools/

How to install an addon extension:

• https://support.mozilla.org/en-US/kb/add-ons-and-extensions-faq?esab=a&s=how+to+install+addon&r=1&as=s#w_how-do-i-install-an-add-on

On computer A which has the emails. Create folder on the desktop for the emails. Right click on folder > ImportExporttool > Export all messages in folder > EML format Save the individual emails as .eml files into the new folder on the desktop. I would suggest saving as .eml files as then there is no chance of overwriting an mbox file.

Move the new desktop folder containg emails over to the desktop on second machine. In thunderbird right click on folder then use the ImportExport tool > Import Messages select the .eml files from the desktop folder and click on Open

First image shows some detail of installing the addon. Second and third image shows exporting and importing eml files