Depends on whether it's an emergency cell or a swarm/supercedure cell. If the bees know in advance that they want to prepare to make a queen, they will start a queen cup and the queen will then lay an egg inside it. The cup can then be expanded into a queen cell. Queen cups/cells are often built just-in-case and may be torn down again before being capped.
In the case of an emergency cell, when the need for a new queen is not recognized by the bees while there is still a laying queen available, they have to refashion a worker cell into a queen cell after the egg has been laid. In that case they prefer a young larva in a newly drawn cell which has soft, pliable wax that is easily reworked into a queen cell.
The workers don't move eggs around from one cell to another.