What we are starting to see is web push notifications becoming fully supported by browsers, and I think it's only a matter of time before iOS and Android start allowing app-like websites to notify users without them having to download a full-fledged app.
It would be interesting to see what this does for the iOS and Android development landscape. They're in a golden age now where every major property needs both a mobile site and a dedicated app on both platforms. But if the mobile site can handle more features that were previously the sole provenance of native apps, that seems likely to shift the landscape.