FTM has a copy and paste feature for facts, but only for self, spouses, siblings, children with each paste. (Select a fact, right-click > copy, then move to various people, select a fact (any fact) and right-click > paste.
You could probably paste such a Fact to each person in the 2,000 person file over a fairly short period of time. Copying and pasting to each person in the 100,000 person file would be a large project.
The Edit > Manage Facts screen provides several things one can do on a global basis (move data from place to description or vice versa, make all private or not, etc), but I don't see a global copy and paste there.
If you were to do a copy and paste to the smaller files, I would attach a source citation to the first fact so that it is copied and pasted at the same time with the facts it is associated with. That gives more fields to use to put relevant info.