The correct answer to this open question is the following.
Unfortunately, you did not specify to what event you are referring to. You did not mention the "exchange" between the two countries.
However, trying to help you, we can assume that you are referring to the result of the Mexican-American War.
If that is the case, we can comment on the following.
What this exchange of ownership from Mexico to the U.S. meant for Mexican citizens living in the Southwest was that they had to decide if they wanted to move out to the Mexican territory after the signing of the Treaty of Guadalupe-Hidalgo to keep the Mexican citizenship, or they wanted to become American citizens.
What happened to Mexicans at the end of the Mexican-American War (1846-1848) who were living in the area affected by the Treaty of Guadalupe-Hidalgo (1848) was the following.
Mexicans who lived in that territories suffered from the pain and destruction caused by the war. There was a considerable loss of property and much damage in towns. However, after the signing of the Guadalupe-Hidalgo Treaty, Mexicans who lived in those territories were allowed to have United States citizenship.
The treaty was signed in February 1848 and Mexico had to give the United States government the states of California, Utah, New Mexico, Arizona, and parts of Wyoming and Colorado.
