In a team mailbox, you can make one person responsible for each conversation. Everyone on the team sees who it's assigned to, and each team mailbox has Unassigned and Assigned to me folders.
Steps
Desktop
Open a conversation in a team mailbox.
Click Assign at the top of the conversation, or press Shift+A.
Choose Assign to me or a teammate. To clear it, choose Unassign.
Unassigned and Assigned to me are in the sidebar, under the team mailbox.
iPhone, iPad & Android
Open a conversation in a team mailbox.
Tap ••• at the bottom of the screen, then tap Assign.
Choose Assign to me or a teammate. To clear it, choose Unassign.
To see Unassigned and Assigned to me, open the menu and tap the team mailbox.
Good to know
A conversation has one assignee at a time. Anyone on the team can assign, reassign or unassign it.
The person you assign gets a notification: in Pulse on desktop, and a push notification on their phone. Assigning yourself doesn't notify anyone.
Archiving, snoozing or replying doesn't change who a conversation is assigned to. Assigned to me keeps showing archived conversations, while Unassigned shows only conversations in the Inbox.
You can only assign one conversation at a time, not a selection.