I think it depends when the key was last activated.. My Win 7 Pro key from like 2011, which has been used on 5 different core machines with at least 6 months in between activation, has allowed me to swap it between multiple machines, inclusive of the Win10 upgrade path. But then when I had to do a quick swap of a HTPC to a different machine, it said the key was in use (different key for diff version of windows). All keys were OEM, not retail. So I think with enough time between activations it just works anyway!
I changed my mobo at the same and SSD too, no issue. The only thing I kept was the frame and video cards (and spare HDD/SDD).
As mentioned above, time between installation may impact and in the worst scenario, call M$ ... normally just typing the numbers is fine. (Choose english, french never yield good results for co workers lol)