If you take a look at the Tooltip of this button then you will see its named "Refresh view" and not "Synchronize" -> the messages are only downloaded from the phone. If you want to move some messages then move them first into the Message-Archive and then you can Upload them back to phone into another folder. rightclick->Upload to phone
Sorry, it's because I had only received messages for testing so it didn't propose me to upload it in the sent messages folder.
Indeed, with a received message, we can.
Thanks !