The less you handle the eggs, the better for them so you don't add oils or bacteria from your skin. Gloves can help.
I like moving her at night so she won't move.
My preferred method of moving to broody quarters is to warm up some fake eggs to about 100 degrees and put them in the new nest. I then move the broody to the new quarters on the fake eggs. If she doesn't freak out, I let her get accustomed a bit. I then collect her incubating eggs and replace the fake ones.
The reason I do the fake ones is if she goes ballistic she won't break the real ones.
Needless to say I have skittish birds.
If you have a calm broody like a silkie, that step probably isn't necessary but I like the idea she isn't off of warm eggs more than a few seconds at a time.
It's nice to have an incubator as a backup in case the move doesn't go well but that has never been a problem for me.