I would guess the iron is in the chips. And that additionally, these boards were soldered with a water soluble flux containing ammonium/zinc chloride. It's probably the steel in the lead frame producing the rust, and the epoxy case of the IC was not adequate to seal it from chloride ions and air getting in there. So tinning can't cover the steel and stop the rust, because it is leaking out from where the pins come out of the epoxy. It's maybe older encasing tech or some failure in processing that damaged the epoxy.
If OP could not buy new chips, I would remove them all, clean the flux residue thoroughly, and dunk them in any acid but HCl to eat away any rust. Rinse in water and dry with alcohol. Dunno how many chips that is. It's certainly a lot of work. But mechanically removing the visible rust probably can't stop the source of the problem. This probably can't either, but it theoretically has a chance... of slowing future problem. If the chips are already acting funny, it's probably too late. It's not just presence of rust you can see, but there is maybe rust in the chip pushing on and breaking the wire bonds.
If you can buy new chips I bet you have no problem in the future, provided you are using halide-free flux. I would wash the FPC thoroughly in hot water and a brush before putting on the new chips. I imagine they would be a horror show on the inside, already, by the time rust is pushed out of the chip.
* scratch all that.
If you have device(s) that still work, I'd clean what I can see with a fiberglass brush, then coat thoroughly with some suitable lacquer/paint to slow any more oxygen from getting into the chip.
I have some PCB I soldered with plumber's flux. Maybe this will happen to them in 30 years.