deleted messages keep coming back

I’m using emClient on one machine and Thunderbird on another (for good reasons not worth explaining).  Messages I delete using emClient still show up in my inbox in Thunderbird, although deleting in Thunderbird deletes in emClient as well.  I suspect emClient is marking messages as deleted locally but not forcing the deletion to the server.  How can I make emClient commit these deletions so Thunderbird sees them?

With Thunderbird and any other email clients (on your mobile phone) connected to that account closed, when you delete a message in eM Client, does it still appear in the Inbox using the web interface for your email provider?