How can I stop receiving new incoming conversations, and still have the time to close the ongoing ones?
As an expert you can decide at any time to stop taking new incoming conversations. To do that, go to your expert profile (1), and click on your status to switch it to OFF (2).
The OFF status will allow you to handle your ongoing conversations without receiving new ones. Once all your conversations are closed, close your discussion panel or click on the "Log out" buttons displayed at the bottom of your expert profile.