Most people who move overseas do so with some sort of goal in mind - missions work, starting a business, teaching at a school, etc. While this is inherently good, and is often the reason for moving overseas at all, these goals can make it easy to enter into the new culture, "guns blazing", ready to make changes. While I could get into all of the negative implications this can have on the effectiveness of the work you do, the relationships built with the natives, and the lack of long-term results you will likely see, I'd like to focus simply on what your attitude regarding entering a new culture teaches your children.
Your children are watching you and are constantly learning from you. In your home culture, you know how to teach them the social norms, rules, customs, and values, but when you enter into a new culture, that set of rules changes and leaves you unsure of how to move forward. In a TED Talk by Julien Bourrelle, he lists three ways an individual can react when they move into a new culture: confront, complain, or conform. When you choose, or subconsciously do any of these, you are subtly teaching your children how they, too, should navigate this new place and culture.
Confront: I could use the word "combat" interchangeably here. This is an attitude of fighting for things to be done "the right way." This attitude might come through in subtle ways such as mumbling, "If they would just do it this way, it would go so much faster!" or in deciding that your family will remain American (or whatever your passport nationality is) in every way possible while living in the different culture. This attitude can either be blatantly against cultural integration, or just subtly unwilling to adapt. Either inherently teaches your children that the passport country's way is the best way, and thus causes them to also be fighters of, instead of learners of, the new culture. It fosters a prideful attitude that says there is only "one right way to do things" and that the people in the new country clearly have it wrong.
Complain: It drives us crazy when our children whine and complain, and yet, sometimes we subconsciously teach them that this is an appropriate attitude toward an unfavorable situation. Living in a new culture can bring out annoyances and push your buttons, so to speak, on a daily basis. Your children are looking to you to learn the acceptable way of responding in those situations. Unfortunately, in my years in East Africa, I heard many parents complain about the local people, customs, or ways of doing things to their children. Often, it was in the form of a nonchalant or joking comment. This does not foster an attitude of acceptance, humility, and respect, but instead enforces a "better than" mentality.
Conform: Or, adapt. This does not necessarily mean that you need to change everything from your clothing to what your family values are, but it does mean learning to live like the people in the new culture do, in some ways. Adapting teaches your children that there is more than one right way to do things, encourages a love for diversity, a respect for all people, and a positive attitude in the midst of potentially frustrating and uncomfortable situations. As they see you working to adapt, accept, and learn the way that things are done in the new place, they too will feel the freedom to integrate into the new culture.
By living in a different culture, your children have the opportunity to truly become lovers of the world - global citizens - in an intimate, life-altering way that would not be nearly as possible if they lived only in their passport country. Don't squelch this opportunity by teaching them to confront or complain. As parents, you are actively teaching your children what their attitude should be toward the new culture, so be vigilant about enforcing a positive attitude; one that encourages conformity and a positive, benefit-of-the-doubt, response when something looks different than what you, or they, are used to. Encourage and display the attitude of one who is a student of the culture, not a teacher there to convert the people to the "right" way of doing things. Foster in your children an attitude of humility by first displaying one yourself.