This is a fairly concise summary of the adoption process in Kaz courtesy of the US Embassy's adoption webpage.
"U.S. citizens hoping to adopt a Kazakhstani child begin this endeavor in the U. S. Once prospective adoptive parents are cleared for adoption in their home state, and have completed processing with the U. S. Citizenship and Immigration Services (Form I-600A), the next step is to register with the Ministry of Foreign Affairs (MFA) to indicate intention to adopt a Kazakhstani orphan. An adoption agent normally assists with this registration by submitting the file of documents ("the dossier") to the Kazakhstani Embassy or Consulate in the U. S. The Kazakhstani Embassy then transfers the dossier to the MFA in Astana for processing, which sends it to the Ministry of Education in Astana for review, which forwards the file to the Guardianship Body in the town where the parents wish to adopt. The length of dossier processing is unpredictable.
Once the dossier has cleared each stage and the parents have cleared a final security check, the Guardianship Body will then issue a letter inviting the parents to travel to Kazakhstan to apply for adoption. This letter is necessary for the parents to obtain a Kazakhstani visa, and after obtaining the visas the parents are free to travel to Kazakhstan to proceed with their petition for adoption."