They do head for places other than the US. When I visited with a Mexican group that sheltered Central Americans, I learned that many headed for Monterrey in Mexico to disappear and look for jobs. Others went to other Central American countries, still others moved within their own countries. And the notion of flying to Lima or Santiago is ridiculous, as that requires money and perhaps visas. The people I observed were frightened, penniless teenagers who were asked to join drug gangs and threatened with death if they refused.
You reminded me of a beautiful song by Roger Whittaker and later covered by Reba:
Just Across the Rio Grande
The lights of Laredo dance on the water
And shine in a young man's eyes
He stands on the border and dreams of Paradise
He's heard crazy stories of how people live
Over in the Promised Land
He heard they eat three meals a day
Just across the Rio Grande
He's got a wife named Maria and a baby named Rose
And one more to feed on the way
Two willing hands that couldn't find work today
He stares at the river and curses the future that he can't understand
He knows his child would have a chance
Just across the Rio Grande
It's only a river that's not that deep or wide
A boy can throw a stone across and reach the other side
It's just some muddy water cuttin' through the land
But a man can make a dream come true
Just across the Rio Grande
The lights of Laredo dance on the water
And shine in a young man's eyes
He stands on the border and stares up at the skies
The same old stars shine so bright over the Promised Land
Sometimes it seems like God must live
Just across the Rio Grande
It's only a river that's not that deep or wide
A boy can throw a stone across and reach the other side
It's just some muddy water cuttin' through the land
But a man can make his dream come true
Just across the Rio Grande
Sometimes it seems like God must live
Just across the Rio Grande