We had one hen who attacked and killed (but didn't eat) a few chicks from other hens. She had chicks of her own at the time but her chicks were three or four weeks old and the others were just days maybe a week old. I can't quite remember exactly now.
Anyways, we didn't know what was killing the chicks (we are free range - and so are the chickens!) until I saw it with my own eyes, she pecked one chick on the head, it ran away, she chased it until she caught it and then repeatedly pecked it until it was dead.
She was sent to the 'home for bad chickens' (aka the barbeque) the next morning.
The odd thing was, when her chicks got to about three months they all started to peck other small chicks, whether she had taught them to do it or it was just a 'nasty chicken gene' thing going on I don't know, so they too were quickly dispatched.
The only chick to survive the onslaught lost an eye and has a concave lump ( a dent?) on her head. She has had three broods of her own and she has never attacked any chicks but she is far and away the most aggressive hen we have. But we have never had another chick killer since.
If you find the culprit - get the BBQ ready. I don't think you'll be able to reason with them.
Moving them with eggs is not a problem, we do it all the time - they lay all over the place here so when they start to sit, we just take them up the coop with their eggs and they don't have any problems. Watch them for a couple of days to make sure they go back to where they are supposed to be after getting off for food and water - sometimes they forget.