If you drag an email to a Gmail account label and it doesn’t appear in the destination label after sometime and is also no longer showing in the original Gmail label, then check in “Show Operations” via the dropdown on the right of Refresh to see if eM Client is still synching the changes and if it is wait till it’s finished.
If eM Client is not synching and closing and reopening eM Client still doesn’t show the moved email, then that usually means the local cache in eM Client needs clearing and starting new.
So you would first go online to your Gmail webmail and look in the original & destination folders / labels and see if the emails are showing there.
If the emails are not showing online in the original or destination Gmail folder / labels “then look in your All Mail folder” either in eM Client or online as you might have accidentally dragged / moved the message in a wrong label where you can then drag / move it to the right label if it is in All Mail.
if you also carnt find the message you dragged / moved in All Mail, then I would suggest to contact offical eM Client support to see what do do.
If they are showing online in webmail but are not showing in eM Client even after refreshing the client a few times or closing and reopening, then “right click on All Mail” under your Gmail account and click “Properties at the bottom”. Then click on the “Repair tab” and click Repair. Once cleared your messages will start synching back to eM Client and will be slower to read till fully cached locally again.
Note: Before running the repair make a manual backup via Menu / Backup” incase you need to restore for any reason. You can see when that’s completed in Show Operations.
Yes, this is with GMail
So dragging basically just adds the destination label and that’s that
My request still stands then: Properly “move” a message from Unread when dragging. 
Yes for Gmail accounts if the email is in the Unread Favourites Global folder and you drag / move it to a Gmail account label you have to still remove the original Tag Label after moving.
However if the email is in other Favourites Global folders such as All Inboxes, you can drag / move it to other Gmail account labels without having to then remove the original tag / label.
So I will test this online in Gmail webmail and see if this also happens with the Unread folder or if it’s only eM Client that behaves like this.
If you could also test it online in your Gmail webmail as well to see what happens.