By default, Gladly will route the oldest Conversations first (by due date) before any new Conversations unless the old Conversations were marked as No Reply Needed. For example, if there are old/overdue Conversations, and new emails waiting for a response, an Agent clicking "Next" will be routed the old/overdue emails in the Inbox first.
If you'd like Agents to always receive work from the "New" queue, we suggest creating a Rule to automatically move old/overdue Conversations to a specific Inbox explicitly made so you can monitor and track these overdue Conversations. Agents assigned to this Inbox can be tasked with following up with these overdue items. You can even set up the Rule to automatically close Conversations if they are overdue by a <configurable> amount of time