That won't happen because of network effects. Nobody wants to edit a site that nobody else reads.
Even if the networks will move over, a lot of links won't. Search engines won't put results from "VeryFreeCompletelyTransparentWiki.com" as a first result/Sidebar, they will put Wikipedia because the large user base can effectively prevent vandalism. Wether that's a good thing or not is up for debate
Networks don't move, individuals do. But individuals tend not to leave a large network for a small one.
Networks can connect. Old and new wikipedia can be connected together to share edits.
> the network of editors moves over to the new Wikipedia