Mozilla has mobile style push notifications for Firefox and you can just leave eg a mail client running in the background using the IMAP IDLE feature for mail.

Mozilla has mobile style push notifications for Firefox and you can just leave eg a mail client running in the background using the IMAP IDLE feature for mail. As long as you restrict that sort of thing to just a couple apps it's fine. More than push notifications the WiFi firmware needs a feature to whitelist IPs that can wake the phone from sleep so you get instant notifications from services.