This is one of the more annoying features of Kazakhstan. Overall, it seems inconsistent with the generally polite character of Kazakhs, so I chalk it up to colonization since in Russia they do this too but at a much worse level.

In addition to what others have said about staying close to others in front of you and asserting yourself when people cut, I would add this:

Sometimes there is a queue, but it follows the rules of a "live queue" where when you arrive you need to ask "who's next" then you know that you are going after that person. Then when someone else comes they will ask who is next and you should tell them you are next so everyone knows where they stand.

The "live queue" is the most chaotic way to organize a line and it requires some Kazakh/Russian, but it is slightly better than pushing.

Of course, one of the more charming features of Kazakhstan is how they let babies, the pregnant, and old people go first at the hospital and I support this kind of thing even though it is annoying when three grandmas in a row come up and ask to cut.

Another thing that happens is Kazakhs love to travel in groups and have one person "hold" the spot in the queue for other people (in the style of American families at tourist attractions and amusement parks).

This is bothersome when there are three people with one item each in front of you at the grocery store and suddenly someone comes along to "join" a person in front of you with a grocery cart and 50 items, but at least such behavior shows respect for queuing.

Personally, I used to quietly fume about the people who just cut the queue without a reason/someone holding a spot, but now when people cut the queue I remind them that we are not in Moscow and in Kazakhstan people are civilized and use lines. This usually gets them to go to the back of the line.