[EXTRA] Implement the moving solutions into CI

Initially we coded our rapid moving solution into a branch (see https://gitlab.com/gitlab-com/gl-infra/infrastructure/-/issues/12121), with the intention of running this script in a long-running tmux-session.

There were some concerns about the maintainability of this solution, as manually-run scripts usually require baby-sitting and manual interventions to correct possible mistakes. Hence, it was suggested that we have the moving solution into CI, ideally in the balancer project. This way it is more familiar to the team and is aligned with our approach of dog-fooding.