Within Teamwork Projects you can move a message to another project on your site.

To do this, go to the Messages area of your project and then select the message you wish to move.

Once you are viewing the message choose the options (three dots) to the right of the message, and select Move Message from the dropdown menu.

In the move message modal, you will be able to select the destination project you want to move the message to.

For more information see: Adding a new message