Hmm, when I've seen an enlarged sac on the neck area, it was a full crop. If the crop isn't emptying, it will develop bacteria and the baby will die. Are they warm enough to digest their food properly? Are you hand feeding them? If so, it's possible you're feeding too much or too often. I don't know about pet finches but house finches are like that. I've had some of my volunteers over feed baby house finches because they were still begging. I showed them the crop area and how engorged it was. Later I showed them how it should look when fed the right amount. I haven't seen your birds so I can't know for sure what's going on. But this is all I can think of right now.
Thanks, it makes sense. Mom bird has been feeding them and the other three are doing great under the same conditions.