There seems to be ongoing confusion on this board regarding what RC is (and is not). RoundCube is an "interface" to the imap server. It is not an MTA, or in itself an IMAP server. It just makes an IMAP connection and interfaces with the server (whether it's Cyrus, or Dovecot or UW-IMAP), just as any client MUA such as Outlook or Thunderbird would.
When you say you moved between two servers, did you just move RC or did the IMAP server (and associated mailboxes) move as well? Or are you using RC to point to another IMAP server (at another name or IP)?
Unless the "sent" mail on the mail server was deleted or moved, it is not "lost", but is perhaps not visible?