If the emails are addressed to you then they will be received by the server and therefore you will see them when you connect to server / download from server.

However, it sounds like most of these emails are 'Junk'.

Make sure your Junk Controls are set up correctly.

See info at these links:

• https://support.mozilla.org/en-US/kb/thunderbird-and-junk-spam-messages
• http://kb.mozillazine.org/Junk_Mail_Controls

Then when you see an email which is Junk, mark it as Junk, so that Thunderbird Junk controls know what to do. Once the Junk Controls have been trained, those emails will still get downloaded but they will go into the Junk Folder. You can periodically check the Junk folder to make sure there are no 'good' emails slipping being misdirected. Then you would mark those emails as 'Not Junk' to train the Junk controls,

re 'The emails keep rolling in, nothing to do with me'

my first response assumes the emails are simply Junk, but what do you mean by 'nothing to do with me'....are they addressed to you?

Please clarify.