I gave up trying to figure out what to do with boxes of extra pokemon from breeding.
Now I save as soon as I put parents in Day Care.
Collect 5 eggs. Check and see if any of them are what I'm looking for or closer to what I'm looking for if I'm IV breeding. If not SR. Refuse the first egg so all the eggs I get are different from the first set. Save. Collect 5 eggs repeat.