In 3rd party chat clients like Pidgin or Adium, operators will follow the chat management link that comes in at the beginning of each chat to send a transfer. The link will take the operator to a special page for management of that specific chat. From that chat management page, they can transfer the chat to another queue or individual operator. They can also send the guest a file, email a transcript of the conversation, tag a chat for follow-up, or block abusive guests.
Example chat in Pidgin:
Chat management page for external chat clients: