There's really no notion of "moving" a project to a working group. I may be splitting hairs...
The relationship between a working group and a specification project is very formal. It is one-to-one. It's not strictly true to say that a specification project is "owned" by a working group, but rather that a specification project works under the supervision (or purview) of a single working group. Specifically, the working group's specification committee has a decision making responsibility (via ballot) regarding whether or not a specification project's releases may produce a final specification.
The relationship between a working group and a "regular old Eclipse project" is less direct. A working group may "express interest" in any number of Eclipse projects. The relationship is many-to-many; there exists, potentially, many working groups that may express interest in any particular project. Strictly speaking, a working group does not "own" any Eclipse projects in that the working group doesn't have any direct influence over any activity that the project engages in. A working group influences projects by having its participant members' employees contribute to the projects in the usual way.
What "express interest" means is really up to the working group. Many working groups have web pages that list the projects that they are interested in, for example.
It's also the case that the developers who have committer status on any of projects that a working group expresses interest typically form the constituency for committer representatives on the working group's committees. So... if the working group were to list Eclipse Equinox as one of its projects, the Equinox committers would join the Eclipse OSGi project committers to form the community of committers who vote on a representative for each of the working group committees.
I'm not trying to suggest that you want to do this; I'm only describing what it means.
Wayne