Expedited capability for migration to your site
At 30k on graduated sites (and 10k on beta sites) grant users the power to suggest migration in a more pro-active way to "their powerhouse".
Example: Say you have >30k rep on site X, but also associate with site Y where you have [not a lot of rep], provide some capability to migrate content from site Y to site X since you're well-equipped with the know-how of what goes/doesn't on site X.
I am a high-rep user on TeX.SETeX.SE, and often find questions on Stack OverflowStack Overflow (SO) that is (La)TeX-specific. As such, I flag these questions, suggesting migration to TeX.SE since the fit is mostly perfect. It just makes sense to move the post to a more suited site.
However, since I'm a lower-rep user on SO, I can only flag a question and suggest TeX.SE as a better location (I assume it's treated similarly on the back-end to a vote-to-close). This flagged is effective < 5% of the time. I think the reason here is that the post ends up in the Close Review queue, which is a particular problem on SO. From a TeX.SE point-of-view, SO almost seems like the garbage-collector of posts since it has so much coming its way, from all walks of life.
I don't see much benefit in leaving formerly on-topic posts on SO rather than migrating it to a site that is tailored to address those issues. This will naturally be the case as more proposals on Area 51Area 51 launch.
In short, if you have interest in a specific field of expertise (read: high-rep on a specific site), you should have a pretty good idea of what is on-topic on that particular site (read: you've hung around long enough to know what's going on) and therefore should know what fits on that site if it lands elsewhere on the network that you might troll on-and-off.
While my suggestion is based on an SO-TeX.SE relationship, it may hold in general. However, I wouldn't know how to address this speculation with some circumstantial evidence from SEDESEDE, say, mainly because there's no easy way to perform cross-network queries/analysis.
This proposal might be consideration for something like an "In-migration review queue" rather than granting "high-rep privileges on a low-rep associated site". This proposal might need some baking...