Is it prescribed to say "In the name of Allah and upon the creed of the Messenger of Allah" when lowering the deceased into their grave, or is it specific to the one directly handling the burial?

It is sunnah for the one who places the deceased in the grave to say: "In the name of Allah and upon the creed of the Messenger of Allah." This is specific to him. As for those attending the burial, they should supplicate for the deceased and ask Allah to grant him steadfastness.