How can one find the names of the descendants of al-Mustazhir Billah, al-Mustarshid, and al-Rashid, especially those whose names may not be well-known or who have migrated, for the purpose of verifying an Abbasid lineage?

It is not the Center's custom to assist researchers due to its preoccupation with answering people, but the following references may be useful to you:

Yusuf ibn Taghribirdi al-Atabaki: In his book "Mawrid al-Latafa fi Man Waliya al-Saltana wa al-Khilafa" (The Source of Grace for Those Who Ruled the Sultanate and Caliphate), in which he discussed several caliphs from the lineage of Al-Mustarshid, such as Al-Mu'tadid, Al-Mutawakkil, Al-Mustakfi, and Al-Hakim. Al-Suyuti: In his book "Tarikh al-Khulafa" (History of the Caliphs) and his book "Nadhmu al-'Iqyan fi A'yan al-A'yan" (The Stringing of Pearls Regarding the Notable Notables). Al-Dhahabi: In "Al-Siyar" (Biographies) and in "Al-Mukhtasar" (The Abridged). Ibn Kathir: In "Al-Bidaya wa al-Nihaya" (The Beginning and the End). Ibn Hajar: In "Al-Durar al-Kamina" (The Hidden Pearls).

You can also use "Al-Maktaba Al-Shamilah" (The Comprehensive Library) or the internet for additional information.