What would I use to transport them?
You should use a short box when transporting quail. The ensures that if spooked during the journey they will not try to escape through the ceiling. You should put a thick layer of shavings in the boxes, never use newspaper, straw or hay.
If quail are scared they will directly upwards ( as they would to evade predators ) and you should be especially careful to not scare them one arrival. To put you quail in their new home, you should place the box into the house or run and leave it open. The birds will come out by themselves. You should not be impatient and force your quail out - otherwise they are more likely to be jumpy.