Attach the copper chest to an arithmetic combinator. Copper * -1 → Iron.

Disconnect everything else from the copper chest, and connect it to the arithmetic combinator output instead. Set the inserters to operate when Iron > 0.
I did this and only the first chest has iron plates left it in, the ...

I have some chests full of iron plates with red wire connected to inserter, and it is all connected to a chest with copper plates in it. The chest with copper in is half full. How do I get the chests with iron plates to empty until they have the same amount in as the chest with copper plates?

See ...