When moving the eggs underneath her, I would say wait till she is almost out of eggs. If you do it too late, there won't be any eggs left and she'll leave. On moving the chicks into the brooder, wait a day to let them get their strength to go into the brooder (don't worry, chicks have a food sac thing in them that allows them to surive for about two days without food/wate. That being said though, always provide food/water for newborn chicks just in case). It also allows them to dry off completely too.
Also, on what Achickenwrangler#1 said about just getting an incubator for the extra eggs. You could, but I've always found it better to use broody hens. Even if the hen decides to get halfway through the new hatching process (My hens usually stay to at least the 3rd or 4th batch of eggs), you can always then put the abandoned eggs in an incubator before they get to cold. This way they would of already been started and your chances of hatching are significantly higher. I do want to mention one thing though. If you do have a broody hen remain broody multiple times, make sure she isn't old. Old hens (5+yrs) get worn out too much while sitting and multiple hatchings can kill them.
Also, on what Achickenwrangler#1 said about just getting an incubator for the extra eggs. You could, but I've always found it better to use broody hens. Even if the hen decides to get halfway through the new hatching process (My hens usually stay to at least the 3rd or 4th batch of eggs), you can always then put the abandoned eggs in an incubator before they get to cold. This way they would of already been started and your chances of hatching are significantly higher. I do want to mention one thing though. If you do have a broody hen remain broody multiple times, make sure she isn't old. Old hens (5+yrs) get worn out too much while sitting and multiple hatchings can kill them.