Let's take a web development environment, where developers checkout a project onto their local machines, work on it, and check in changes to development.
These changes are further tested on development and moved live on a regular schedule (eg weekly, monthly, etc.).
Is it possible to have an auto-moveup of the latest tagged version (and not the latest checkin, as that might not be 100% stable), for example 8AM on Monday mornings, either using a script or a built-in feature of the VCS?
Certainly, but the exact product may be dependent upon the VCS you are using.
What you might want to do, is have a a few different branches, and migrate up as you progress. E.g., Development -> Stable-Dev -> Beta -> Production. You can then simply auto-update to the latest version of Stable-Dev and Beta for your testers, and always be able to deploy a new Production version at the drop of a hat.