What is the proper process for updating to a newer release of ABS?

That largely depends on what version you upgrading from. For minor updates you can install both client and server over your current installation. Depending on the changes between the version you may need to disable and re-enable your clients. For major updated you need to un-install your server, deleting it's database and restart your server setup from scratch.

The detailed instructions for each release are usually contained in the release notes
http://trac.addressbookserver.com/trac/wiki/ReleaseNotes

Instruction on how to un-install and re-install are available there as well
http://trac.addressbookserver.com/trac/wiki/Upgrade

Please let me know if you have any further questions

I have the Dec 13 beta installed and want to update to the current Dec 20 version

Just install over the top. Nothing special required there