I use the export function in Outlook and categorise my emails so only backing up the colour/s I want. If you don't know what this colour means then in Outlook right click an email and on the list of options click categorize and you will see the list of colours you can choose from. Once I have them backed up I select to view as Categories and unmark their colour/colours.
If you do use Outlook it varies by the version you are using to get exporting done. Each time I give a name to the backup .pst that allows me to recover particular dates and when you want those files back in you just import that file. If you want fine tuned help on it PM me and I can try to respond to your particular version needs. Outlook 2016/2013 or 2010 I am currently very familiar with but I can go back to my notes for older versions.
Thunderbird is a little different:
Click "Save as" then chose the folder you want to save to (can create a new one or use one already created) You can name the folder with the date or whatever allows you to remember eg 20180321.Emails or WeddingInvite.Emails
You chose the emails you want to save (highlight them, can choose one, multiple files, or all of them)
Your separate emails will be saved to the folder you selected.
If you need to open one choose "File" > "Open" > "Saved Message..." (it will open last folder saved to, if you want a different folder just use the side menu on the folder view to select the folder you want) then select the email you want and it will open for you.
If you want to bulk save for offline use (first time) then
choose "File" > "Offline" >Click "Work Offline"
then choose "File" > "Offline" "Download/Sync Now..." > select what you want to save eg Mail messages then click "Select..." and in the next box choose email account/accounts you want to download/sync click "Ok"
then on the box you return to click "OK". This will create your local files
Working in offline mode stops any issues from new mail messages arriving while doing this. After you have done these steps go back to "File" > "Offline" > Click the "Work Offline" which should have a tick beside it to return to online mode.
If you want to update these at any time just click the "Download/Sync Now..." option.
The local folder directory for this is the default one and can be found in %AppData%\Thunderbird\Profiles\ or by clicking "Help" > "Troubleshooting Information" and look for Profile section in the list and click "Open Folder".
In the list if you use POP you will se PopMail and if you have IMAP you will see ImapMail. In these folders you will see a list of your accounts as folders and if you open one you will see each folder of your mail eg Inbox there named as eg in the case of the inbox Inbox.msf.
This profile directory can be changed to another folder of your choice if desired.