When questions are migrated to another site, do any flags on that question (and any of its answers) get migrated to the other site?
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From memory, no they don't. I say that because "we" regularly migrate posts from SO to MSO after they get flagged, and I don't recall having to clear any flags on MSO. It makes sense really - quite likely a migrated posts is flagged for migration, and it will soon get attention on the other site.
My ♦ only extends to SO and MSO, so it might be different in other scenarios, but I doubt it.
No. Flags, open and close votes are not migrated. The question, its up/down votes, the answers and their up/down votes, and the comments with their upvotes are copied. Everything else is a clean slate.
It will get flagged quickly enough if there's a problem.