That is what I'm suggesting, but I've just done my experiment with Thunderbird and things don't work as expected.
If you set up a message filter in Thunderbird and it remains up and running and a message arrives that you tell Tbird to move to one of your IMAP folders the message does, indeed end up in that IMAP folder on the web interface to the same e-mail account.
However, if you then shut Tbird down, and send a subsequent message that meets the filter criteria, it does not end up in the correct IMAP folder when received at the web interface. This tells you that the Tbird message filter is being run on the client side, and only the client side, so that the filtering will not be the same across devices unless you were to have Tbird running all the time.
The next time I fire up Tbird and it does it's first send/receive it recognizes those messages and filters them appropriately, and the messages are moved to the correct folder locations on the web interface as well.
This all tells me that Tbird is not setting up remote filtering in concert with the server, but is setting up local filtering (though I'm using IMAP folders as my move-to destinations), and that unless Tbird has received the e-mail, run it's filters, and moved the messages they will sit in the inbox elsewhere until it does. This is clearly not what is wanted.
I actually want to thank you very much for forcing me (so to speak) to do this experiment to confirm the precise interactions between the e-mail client and the IMAP server. I have to presume that there exists an e-mail client that would work in concert with an IMAP server to create the filters on the server side, but that client is not Thunderbird. I have always done what you are attempting to do and set up my folders and filters entirely on the server side and let it do the filtering and moving of messages to the appropriate folders. When this is done the results are always automatically reflected on every device that accesses the account via IMAP the moment it does it's next sync (send/receive) with the IMAP email server. This behavior is one of the major things one hopes to get out of an IMAP access arrangement.