Speaking from personal experience, I was raised both in Bulgaria and America for half of my life and has definitely given me two different perspectives about life and how to live with a certain culture of people and how to approach people, ideas and functions with an open ended mind.
No one has an unchanging culture, and as such, cultures merge, and blend. Certainly children will experience this regardless of their upbringing, however it might not be such a shock if they are raised to be culturally aware and to be part of society at large. Often immigrants want to raise their children with their own traditional values. This is not fair to the children who then grow up and live in their new nation, as they are completely unprepared.