How to amend a return to remove a dependent so the intended parent can claim them?

I recently submitted my taxes and they were accepted. One of my children were not meant to be claimed on my taxes, her father who we live with, is to claim her. I've seen several recommendations floating around, but I'm unsure of what to do. I was going to wait until AFTER I receive the first taxes to come back, to file an amend to remove the dependent. 

Should he wait to file his return via mail (including the same dependent I will make the amend to remove) until after I submit my amendment? Or would it be best for us to mail out my amended tax return (removing the dependent) & his tax return (to ADD that dependent) at the same time to avoid him having to wait until after my amendment is processed? I want to make sure we were able to file how we initially planned to.

I saw something also stating that even if he sends his tax return in with the same dependent I did my amendment to remove, he still cannot claim her because she was previously claimed, even though the return was amended to remove her. Does anyone know if this is true?