I have a few questions. How many zebra finches do you have ?
How old are these baby finches? I'm guessing they didn't all hatch
at the same time. As far as advice, I'm going to hope that someone
else here on this forum can offer you some advice. My experience
is with a single dwarf parrot and a single male cockatiel. I've never
taken care of baby birds, but I'm hoping there is someone on this
forum who reads your text and can offer you some helpful advice.
One common reason for feather loss is feather mites or lice. That would require a quick skin test by a vet. There are sprays out on the market to treat ectoparasites but they are not recommended for very young birds.
The one that is on the bottom of the cage may have some illness or birth defect, perhaps even metabolic bone disease where the bones are not forming well enough to support his/her weight. Not sure what kind of diet you've been providing to the parents. Extra calcium is generally needed during breeding and caring for nestlings. Or it could be a simple case of a young bird in a group not thriving. I've seen it too many times.
I know avian vets are expensive but I think that's what you're going to need to figure out why the young ones are not completely healthy. It may be a simple diet change is needed or it may be an illness. But it's hard to tell online like this.