We do to a degree, depending on the amount of more pressing matters that there are to attend to. I think that I can speak for most of us when I say that we really enjoy the periods of time when the flag count is low and our abilities can be put to use helping high reputation users with janitorial tasks.
During that time, the information available in our moderator dashboard becomes much less interesting than the information that is available in the 10k+ tools section. This also gives us time to help look for and deal with meta tags that have been re-introduced, dig up good candidates for synonyms, spend some time looking at the more popular tag wikis and everything else that outwardly makes the site more useful.
We have, on several occasions had conferences via chat with moderators on other sites to see if any of our famous but now off topic questions would be a good fit. Some of them were migrated, but the majority of them remain locked on Stack Overflow with a banner added by a moderator that looks something like this:
This question is available for historical purposes because it is an important part of the history of Stack Overflow. Please don't misinterpret this question as a good example of what should be asked, and be sure to read our FAQ.
More recent questions closed as off topic and not migrated by the community are usually handled that way because the question does not meet the quality that we want to send to a sister site, or we know that it would duplicate many other questions on the target site. These are naturally deleted over time by users with 10k or more reputation.
Going back to our more 'active' moderation when the bulk of our time is spent servicing flags, we're more likely to intervene in order to stop a bad migration than we are to cast a single vote to send a question to a sister site. This is especially the case when dealing with our 'greatest hits' anthology. The fact of the matter remains, newer SE 2.0 sites simply don't want questions that are several years old with accepted answers. Additionally, just like on Stack Overflow, those questions might serve as a bad example of something to ask.
At this point, I think we finally have a consensus on most of the old (and now not constructive) questions that can be found on Stack Overflow. This is good, because there really are more more productive things we could be doing when the pressure in the moderation queue gets closer to atmospheric conditions :)
To that end, I think I can say we've really done all that we can do regarding the types of questions that you mentioned, at least at this point.