If I move a mail from Inbox to an IMAP folder, the message shows up in the new folder in Mail.app, but if I log into my mail account on Roundcube, the mail is still in the Inbox folder displayed with gray text.
How can I move mails correctly like in Thunderbird or any other email client?
The IMAP server is Dovecot, SSL enabled.
iMac, OS X El Capitan (10.11.5)
Illustration: https://www.dropbox.com/sh/4e7bxwmg20kfp8y/AACL3mJMxoKprT5ndZiAr719a?dl=0
(link updated, anyone can view it [just close the dropbox login popup])
That page requires a login, presumably yours, so it is useless to anyone interested in looking at it.
I can't be sure from the images you posted, so let me ask: is the Cron mailbox inside Mail under "Mailboxes" or "On my Mac"?
The folders and mails are stored on the remote server under the same account. So yes, it's under Mailboxes.
Update: added a screenshot of how normal mails look like in Roundcube (black text instead of gray).
Mail.app IMAP problems when moving mails to different folders
