A push update is what keeps a pass alive after it is added. When something about the member changes, a points balance, a tier, an active offer, the issuing platform pushes the change, and the pass in the wallet refreshes to match. The member never reinstalls or refreshes anything; the pass is simply current the next time they look.
The mechanics differ per wallet. On Apple Wallet, the platform sends a push notification scoped to the pass, the device fetches the new version from the pass's web service, and swaps it in place; this depends on the device registration created when the pass was added. On Google Wallet, the platform updates the pass object through the API and Google propagates it to every wallet holding it. A wallet platform drives both from one change.
Updates can be silent or visible. A silent update just changes the pass. A visible one pairs the change with a change message, so the wallet announces it on the lock screen: the pass itself becomes a messaging channel, without an app and without a notification permission prompt.
When evaluating a platform, the end-to-end path is what to verify: make a points change, a tier change and a message propagate to a real device, and check whether the platform can show you delivery, not just that an update was sent.