Exporting an email sub-folder from plusnet webmail

[#1]

I'm trying to gradually wean myself off plusnet email system to a more independent email provider like gmail.  One problem I have hit is that I have quite a few past emails stored in a subfolder of INBOX. (There are 1168 at the moment). Because these are business emails, I would very much like to export them and store them preferably on my new email account, or maybe in Thunderbird.
Unhappily, I cannot find any way of exporting from plusnet webmail. Is there a way?

[#2]

Configure Thunderbird (or another email client) to use either a) IMAP and copy emails to another system or b) to use POP3 and download them all to your PC?

[#3]

Thank you.  I can see and process plusnet emails in Thunderbird, but cannot actually see the subfolder that I want to copy/export.  I can see it on my tablet email program (Android) which is odd.
It may have to do with how I set up plusnet emails in Windows on Thunderbird some time ago. I used POP mailserver, defined as mail.plus.net, not IMAP and there is another oddity, that an outgoing server of relay.plus.net will not work either, it needs to be smtp.googlemail.com
I may try changing my plusnet setting to IMAP, but am a bit concerned about the fact that I need to access plusnet emails from all over the place, using plusnets webmail service. Emails must not be removed from plusnet server in other words, just copied. I may need more detailed advice I'm afraid.
One other thing, might be relevant, might not be. I have had my plusnet account downgraded to email only, called  'PlusNet Residential Generic Services Product', my broadband is now supplied by BT.

[#4]

As PWatson has said I think its a case of collecting the mails using an email client of your choice rather than exporting them 
Edit/ I use outlook and the standard clients for IPad and Android tablets and smart phones. Then at least if you loose the will or the knack to log onto PNs website the emails will still arrive!

[#5]

Quote from: gbhall

I may try changing my plusnet setting to IMAP, but am a bit concerned about the fact that I need to access plusnet emails from all over the place, using plusnets webmail service. Emails must not be removed from plusnet server in other words, just copied.

That's the benefit of IMAP - Emails are retained on the server and are visible on all devices and, if you choose to delete an email, it is removed on all clients.  IMAP also supports multiple folders whereas POP3 only allows access to your Inbox.
Your Android device and PN Webmail are both using IMAP to access your mailbox.
Try searching for 'imap vs pop3' for some background reading.