Pros - none.
I know it seems to make sense that flushing would "take out" nutrients, but let me ask you this: how?
The plant won't break them down. They won't evaporate. They're definitely not going back down through the roots. I feel like a broken record on this and I apologize, but there is nothing you can do to remove basic nutrients that are now a part of your plant. Check out the actual science, flushing is a waste and harmful to your end product. Peace my friend!