Apps on mobile devices are generally unable to maintain network connections while not in focus.

Is this due to the nature of iOS and Android and them not allowing apps to do that, or is there another limitation at play here?

iOS does not allow apps to be kept alive in background at all. On Android you can (mostly) keep an app open in background, but you should not, as it affects battery life.

Of course, another issue is that no device is always online, so you can't base the system entirely on devices communicating with each other peer-to-peer. You want to at least have supernodes. When you do, it effectively becomes a "dynamic federated network".