How to disable Apple Push Notification Service (apsd) which is trying to bypass firewalls
I figured out that I needed to enter the following in the terminal:

sudo launchctl unload -w /System/Library/LaunchDaemons/com.apple.apsd.plist


I did this already, and it seemed to fix the issue, but would love to know if this is the advisable way to solve the problem.

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Also, in case you want to enable the process afterwards, you can use this:

sudo launchctl load -w /System/Library/LaunchDaemons/com.apple.apsd.plist


sources:

https://developer.apple.com/library/mac/documentation/Darwin/Reference/ManPages/man1/launchctl.1.html

http://www.cyberciti.biz/faq/disabling-unnecessary-mac-osx-services/