Hi. I breed platties, so maybe my experience can help. There is a ton of info all over the web about platties too, since they are so easy to breed.
A platy is pregnant if it's a girl, it's with a male, and it is fat. So, most platties come to your home pregnant for the store or become pregnant if there's a male around. Your female will probably chug out fry for a long time!
Yes platties can definitely get stressed from moving. Is she in the nursery right now? I do not put my platties in nurseries because that seems to make them hurt themselves. If the mom has already dropped fry, I wouldn't put her in the box. Mine drop once every 3-4 weeks.
She probably had more than 4 fry. And the reason the dad may be acting calmly is because he's digesting!
Platties and other fish love to eat fry. If you don't have a lot of hiding places for the fry, or a fry-only tank, move the fry to a breeder box or net for a while until they get a little bigger and have more strength to escape hungry mouths.
Platties have no parental instinct, so I don't think the male is sad because of the fry. Just check your water and make sure it's really nice and clean. Watch him to make sure he isn't sick or if other fish are not bullying him.
If your water levels are as pristine as you can get it, the fry will be fine. They are best left alone with frequent feedings and clean water, I think. They are very hardy. I don't really take care of mine anymore
They give birth in a tank full of moss, I remove the mom, and put the fry in a fry tank. I feed them and clean up but leave them alone otherwise.
You can except to lose a few fry during the first few weeks by natural culling of the ones that are sick or deformed or too slow to not be eaten.
So besides feeding and cleaning, and finding a place to keep the fry safe, raising fry is very simple!
Do you have something to feed them? Their yolk sac with be absorbed soon and then they will start looking around for food. Finely crushed flake food is fine, but there are better foods out there. I would feed 4 times a day if they are in a breeder box, and clean up when they are done eating. Putting some moss or another plant in the breeder box can give the fry some shade and give them stuff to nibble on inbetween meals.