Once you enable the Round Robin feature within your Shared Inboxes, the members  in the shared inbox will be assigned messages that are distributed evenly.

Let's say you have a Shared Inbox with 3 users
The first message sent to the Shared Inbox will be assigned to user 1 on the list and the next messages will be assigned to members in chronological order after that. The next incoming conversation will be assigned to user 2
and the next new message will be assigned to user 3. 

To enable Round Robin for your Shared inboxes, go to Settings > Inboxes, select the inbox you want to enable this feature for and toggle the feature On.

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