I have set my iPhone to show notifications of the number of drafts within a workplace called “actions”. If I delete a draft within this workplace on my MacBook, the notification on my phone won’t refresh automatically until I open the app.
Apps can’t run in the background arbitrarily on iOS, the system removes apps from memory to allow space for other processes. If an app has recently been running, it is allow to run in the background for brief periods, but will eventually get shut down and so Drafts can’t check for updates from sync when it’s not running.
You will notice if you have recently launched Drafts on a device, updates from other devices may immediately update the badge because Drafts is still running in the background and has not be shutdown by the system yet.
Drafts does participate in the background refresh system, which allows apps to ask for time to update periodically, when the system will launch the app silently in the background and let it refresh its data. Assuming you have not disabled background refresh in iOS Settings, Drafts will get launched periodically to check for updates and the badge will get adjusted when this happens. It’s up to iOS to schedule these updates, and their frequency varies with a lot of factors - mostly around power management (battery level, network/wifi connectivity, etc).
Drafts also participates in iCloud’s notification system, which sends push notifications to other devices when changes are made which helps make these background refreshes more effective…these notifications are also subject to iOS scheduling, and will not get delivered in some circumstances.
Long winded, but mostly to indicate that Drafts does what it can to be refreshed, within the limits of the OS.
NOTE: Side note, many people are not aware of…if you force quit an app in the multitasking tray, it opts that app out of background refresh for a time. So don’t do that!