At the Air Astana check-in desk in the airport, ask to check your baggage directly to your final destination. If the airport's customs rules, technical possibilities of airport and other conditions allow it, your luggage will be processed to the final destination. The baggage tag will indicate the airport to which the baggage is checked in. You will find your suitcase on the baggage belt when you arrive at your final destination. If you want to get your baggage in the city of your transfer, tell the staff member at the counter.
Sometimes, you must check-in separately for all flights in your itinerary because customs regulations may require it if you are flying on different airlines or if the transfer time is longer than 24 hours. In this case you will be warned about this during registration.
Online check-in is available by the link.