Generally when you have breeding pairs, you need to only have one pair in a cage. If you have a very large aviary that you can walk in, sometimes you can let the pairs breed. But more often they end up fighting over territory and females, and raid each other’s nests. You also have no control over which birds are breeding. A lot of times, one male will become dominant and will breed with multiple females. This does not work out because each female needs a mate to help with the eggs and chicks. You also have no way of knowing which chicks are related. You should never allow related birds to breed. You should be putting all weaned chicks in a different cage, or find them new homes. If the weaned chicks are in the aviary with the parents, you need to remove them while you can still tell which birds are which. The chicks can mature quickly, and before you know it they are breeding with each other or with the parents. So chicks need to be kept separated by sex.
As to your situation, it’s hard to say what is going on. Eggs are laid daily or every other day, so each chick is a day or two younger than the one before it. With 5 chicks, the youngest could be 5-10 days younger than the oldest chick. This explains their size and feather difference. The parents may have kicked them out of the nest when the others left it. Parents are eager to nest again, so as soon as chicks leave the nest, the nest should be removed. You should not allow a pair to nest over and over without breaks in between. They would not do this in the wild, because breeding season ends. In captivity, it is your responsibility to control how often they breed. The parents should be rested for 4-6 months between clutches. The nests should be removed during this time – they do not need a nest other than for eggs and chicks. It’s also possible the other pair raided the nest and threw the chicks out. In many cases one pair will kill the chicks of another pair. This is because their instinct is to continue their genetics. The chicks are not weaned if they are not feathered. It would have been best to return them to the nest to see if the parents would go back to feeding them. You can try to hand feed them. They will not survive without being fed by the parents or by you. Hopefully you have been able to feed them or the parents have fed them. I know it’s hard to feel helpless when little chicks are concerned.
I would recommend taking down the nests and not breeding your birds as long as you have more than one pair in a cage. Finches will live fairly peacefully in flocks as long as you do not give them nests, or allow nesting anywhere in the aviary. Any eggs should always be taken out and discarded. Otherwise, if you want to keep breeding, I recommend only having one pair in each cage. And again, control how often they breed by removing the nests after each clutch. In the wild, they would have 2 clutches per year, during breeding season, at the very most. But one clutch per year is more common.
Thank you for asking Lafeber,