Puzzling. Messages of a POP account are stored locally, not requiring any internet access. Are both accounts POP?

I created a new Thunderbird IMAP email identity in an attempt to fix the problem of my email no longer working due to overloaded webmail storage. I have now changed both identities to POP.

iinet no longer offer an email service and from what I have seen of the transition to the messaging company is a rolling disaster where the depth is about the only thing that varies. Insufficient information presented to folk that do not understand the technical terms used appears to have been the order of the day. https://help.iinet.net.au/iinet-email-messaging-company

Have you paid the messaging company for the continued use of your iinet email address as is required per the link?

How about you post the troubleshooting information shown in Thunderbird so we can see exactly what you have configured. Help menu > troubleshooting information. Use the copy to clipboard button to place the stuff on the clip[board and then paste the information into a reply here on the forum. I have my doubts you actually have a POP account configured in Thunderbird to get mail from and that is going to really confuse this issue.

I do not know what your comment about an overloaded webmail is about, but IMAP only can be a duplicate of what is on the server as the server is the same one webmail is stored on I am totally lost with this.

Matt, I have had no problems with The Messaging Company until recently. I have configured Thunderbird for POP, as recommended by The Messaging Company. I have deleted emails from the web server, so that I am now well within the server storage limit. I can send emails both from Thunderbird and from webmail. Incoming emails appear in webmail, but do not download to Thunderbird. I am awaiting further advice from The Messaging Company, which currently has systemic issues with its Iinet email addresses. If David is right, I should be able to move emails within Thunderbird without connecting to the web server.

Just to add to Matt's comment: if the account is truly POP, there is no connection to server, but the question is whether that is the case.

The problem seems to have resulted from a Firefox glitch. I fixed another problem by disabling two extensions: FileLink Provider for Dropbox and ImportExportTools NG. That problem remained 'fixed' even after I had re-enabled those extensions. After doing that, I can now move emails between my accounts.