Many of you are already married or are planning to marry a woman from Ukraine. There is a requirement in Ukraine that when a Ukrainian citizen becomes a permanent resident of another country, that person must have a special stamp allowing them permission to live abroad. Without this stamp, you may be denied the ability to leave Ukraine if you return.
Much has been written on this topic. It can be a problem for a green card holder who exited Ukraine through Warsaw following the K-1 interview, obtained a green card in U.S., and then returned to Ukraine to visit her family after getting a green card. When they attempt to leave Ukraine, they are denied the right to leave until they get this exit permission stamp. You don't always get caught(and usually, from most accounts, you won't), but in the present climate of increased security, it's not worth taking a chance.
Last fall I looked into obtaining this stamp which can be obtained through the Ukrainian embassy in the U.S. However, I stopped the process because I was concerned about several issues regarding present and future ownership of property in Ukraine. The issue arose again last week regarding my child's 15 y.o. son. A child is normally carried on his mother's passport until he is 16 at which time he must get his own passport in order to leave Ukraine. Since he is turning 16 next month, my wife inquired in person last week at the Embassy of Ukraine in Washington. She was told that her child could only get a Ukrainian passport (as opposed to the Ukraine international passport) until he is 18. He can be carried on the mother's Ukraine international passport until 18. However, since my wife has a green card, she was advised that she would be denied exit from the consular authorities in Kiev, but would probably not be stopped if she used her Ukrainian passport to enter Russia.
Moral of the story: 1) after you get your green card, you should enter and exit Ukraine through Russia so your wife can use her Ukrainian (as opposed to the international passport with the I-551 stamp (green card) and avoid the possibility of being stopped. 2) get your child a Ukrainian passport while in Ukraine if over 16 so he can sneak out through Russia. 3) or alternatively, you can get the stamp, but this may impact or affect any property owned in Ukraine or don't get the stamp and enter/exit through Russia using the Ukrainian passport or 4) just don't visit Ukraine until your wife and children become U.S. citizens with a U.S. Passport. It sure would help things if Ukraine would abandon its opposition to dual citizenship!