I am not aware that they will move eggs, there isn’t really any need for them to do this activity. They can raise a queen anywhere available larva is located.
If you are trying to introduce a new queen and queen cells are present or a laying queen in present they will ignore the caged queen or ball the cage and eventually kill her or let her starve.
Prior to introducing a caged queen destroy any queen cells present and make very sure you don’t have a laying queen. wait a few hours then introduce your store bought queen. Good luck
The Busy Bee teaches two lessons: One is not to be idle and the other is not to get stung.