Bird man, if you are trying to get as many as you can from a hen then you would be best to let her lay, move 1st round to feeders, let her sit and hatch 2nd round, when she lays 3rd round move them again to feeders and let her sit next round etc. This way you will easily get 5/6 rounds from her in 1 season but doing this will reduce the breeding life of the hen as they only carry a limited amount of eggs and oncw they are gone they are gone.
Eggs can easily be stored for weeks as has been said above so its best to pair your stock up a few days before you pair your feeders so that the feeders are laying after the stock to enable you to put the collected eggs under them.
Phil