If you want to talk about American Civil War-related migration patterns that had a lasting effect on the U.S. social, political, and (obviously) racial landscape, look no further than the African American migration north. Although many African American families and individuals did remain south of the Mason Dixon after the Civil War, considerable portions of African American communities within the former Confederacy moved to northern cities like Detroit, Chicago, New York, Philadelphia, etc. This massive influx of African American labor radically altered the workforce, not to mention neighborhoods and communities that were suddenly split along lines of color. African American culture bled into the fabric of these communities, and infused them with radical new ideas regarding music, art, theater, and literature. Truly, you cannot overstate the impact this shift in population had on the composition of the country from 1865 onward. It also homogenized "white" culture in these northern (and often western) cities by eliminating previous lines of distinction between Irish, Italian, Protestant, or Catholic: whether one was black or white became the new deciding factor of "otherness" within male circles.

[Sources - Kevin Boyle, 'Arc of Justice'; Linda Gordon, 'The Great Arizona Orphan Abduction'; Bruce Catton, 'A Stillness At Appomattox']

I think people are Miss-answering the OP's question. I see his question asking about people leaving both the north (to avoid the draft )and the south (to avoid actual combat and the draft)? Perhaps also the effect on peoples arriving in the USA from abroad or delaying emigration to the USA until things cooled down?

It was the other way around. Rather than hundreds of thousands of people fleeing America, there were hundreds of thousands who immigrated into the Union during the war. Lincoln passed the Homestead act which gave free land to immigrants. He also passed the Immigration Act which further encouraged immigration.
I don't know the numbers of emigrants but don't think it was significant. Most able men were either engaged in war, industry, or farming and there was no major reason to leave.

You might check out "Routes of War: The World of Movement in the Confederate South," by Yael Sternhell, who was a student of James McPherson at Princeton.

I heard that some white Confederates fled to South America after the war. Is there any truth to this and how widespread was it?