When you are upgrading the second arrow for your archers, the lumber mill seems to forget to let any other mills know, that it already has it handled.
As you can see, the right-most lumber mill in the picture, is beginning to research the second arrow upgrade for my archers.
Clicking on the left mill shows that the second arrow upgrade icon is still there. In fact, any additional mills you have will still display the icon even though it is being researched already.
But that's not all. You can begin to research this same upgrade in any of these other mills, and the game will try to research it, again. The first upgrade will complete and any other 2nd arrow upgrades you have started will continue to research.
So what happens...
The original upgrade will cause the game to tell you that a 'job's done'. Any other upgrades you have started will cause nothing to happen at all.
The job will not acknowledge anything actually being finished. Also, you will receive no resources back for allowing this second, same upgrade
to finish, nor will you get any extra damage added onto your archers.
-Thanks to Stryker-17