Within Teamwork, you can move messages to another project on your site.
Go to the Messages area of your project and select the message you wish to move.
Once you are viewing the individual message, click the options button (three dots) to the right of the message and select Move Message from the dropdown menu.
In the move message modal, you can select the destination project for the message.
Note: Only project admins and site admins can move messages between projects. Project admins can only move messages to other projects where they are a project admin.
For more information, see: Adding a New Message