Thunderbird has no provisions for blocking anything. It downloads messages that are in the Inbox on your providers server. If your other email client downloads them and does not leave them on the server for Thunderbird then you have your answer.

Are you using POP or IMAP?

Thanks Airmail. It is POP. Let me reword what I said: Most emails on Tbird are blocked. Then after I allow them to be read THEN they are available on gmail...

I have no idea what that last post means. Is gmail your provider? Then they should be available first on the gmail server. That would be where Thunderbird downloads them from.