Thanks for the info about Gmail pushing to devices.
I observe that my various devices or different user accounts receive these pushes at different times. Some of them are delayed in receiving Gmail for so long that it remains hard to stay up to date in a flowing conversation, unless I manually poll the server. Similarly, if someone emails me asking for something I may not receive that email on one of my devices without delay.
Currently I'm using a PC, Phone, and Tablet all logged into the same Gmail account.
I received a new email on the PC (Windows 10, Chrome) 20 mins ago. I see it in my Inbox in Chrome
The same email hit my Tablet 10 (Nexus 7, Android 4.3) 10 minutes ago, so at least 10 minutes after it was sent to me. I see a Notification in Android. The Gmail app is open in the background.
The email has not yet hit my Phone (Nexus 6, Android 7.0) ... so it's at least 20 minutes "late" in my seeing it.
I haven't opened the email on any device and haven't force-polled the app on any device. The Gmail app isn't opened on the phone but is set to Sync automatically on this account. Eventually I expect to see the notification for the email appear in Android Notifications.