When a question is migrated to another site, a few things then (usually) need to be done to it:
- It must be edited to have enough background information, and be reasonably worded, for the new site.
- It needs retagging.
- Comments should be deleted as obsolete (usually the ones about topicality and migration).
Of those three, the target site's denizens can perform the last two — retagging and comment deletion. In fact, even if someone from the migrating site does those, someone from the target site should check them. But editing for background information is best done by someone from the migrating site. Things that are completely clear on the migrating site, that any user there will understand, may be completely opaque to most users on the target site. (Example.) It is therefore in the target site's best interests for someone from the migrating site to edit the question on arrival.
I propose that, when a diamond mod effects a migration, he have five minutes' edit privileges on that question only.
Even if, as hjpotter92 suggested in a comment on this question, the mod should edit before migration, such temporary privileges would be very useful in case he didn't do so.