If your support team has Jack, Alan and Leela, Create Shared Labels “Task: Jack”, “Task: Alan” and “Task: Leela”. Also create a label “Status: Done” to mark closed queries.
Add label “Task: Alan” to an email query to assign it to Alan. Alan can add the label “Status: Done” when the query is closed.
Use Shared Notes to have internal conversations about Support Tickets.
Simply search your Gmail to find open or closed queries, assigned to any team-mate.
For example, to find open queries assigned to Jack, search for "label:task: jack !label:status: done"