Does "Magic" answer your question? Welcome to Apple, if you want documentation you are dealing with the wrong vendor. Most of what you are wanting will be dealing with APNs.
From the macOS perspective:
- APNs is what spurns the process to handle configuration profiles. Once a device has been made aware the MDM has something for it via APNs, the device triggers the connection to the MDM to install the profile. In my experience this is generally less than 5-10 minutes for macOS. In many cases devices will start to react in under a minute from the MDM broadcasting the configuration profile.
- A configuration profile installing, and a configuration profile taking effect are different. Configuration profiles install in /Library/Managed Preferences and are also visible in System Preferences > Profiles. Some configuration profiles need a reboot or a MDM framework update to start working which can take time.
- Sudo JAMF Manage would be the command you are looking, though this command does not do as much as it used to.
If no devices are getting configuration profiles: Make sure APNS is working. Check certificates and network traffic.
If some devices are not getting configuration profiles: Check JAMF's configuration policies, the scopes and triggers to make sure they are right. You will always have some devices being dumb, you can check the inventory record for configuration profile install status.
Most of this is generally the same for iOS, though you have much less visibility on the OS side over what is going on.
Apple developer may have some of the information you are wanting. What's New in the Apple Push Notification Service - WWDC16 - Videos - Apple Developer
