Comment by AgentME

The Gmail API works nicely like this. There's a history.list endpoint where you can see the recent history of message additions and removals, you can query for just the history that's taken place since a specific event's historyId, and you can subscribe to push notifications (that can be delivered by webhook) which just tell you when there are new history events, and you're expected to hit the history.list endpoint to see what's new. Some dropped push notifications aren't a big deal.