The OS provides a messaging service. Applications can request differing levels of latency guarantee. The default on Android is to deliver immediately unless the device is off or sleeping, in which case it'll be queued for up to 4 weeks or until the device wakes up, whichever comes first. There is also throttling functionality, where an app that sends too many messages in a short period of time will find that its messages are only delivered when the device radio has to wake up for other non-throttled messages.

With iOS, Apple provides a service that event sources push their messages to, and a single TCP connection is maintained with the device regardless of the number of event sources. Google has something similar, if not identical.