For example, in 1541 Coronado entered present-day New Mexico searching for "cities of gold," and one of his first actions on meeting the natives was to burn 100s of them alive in their dwellings, for not handing over suspected horse thieves. That is how Spain conquered the natives of the present US Southwest--not nicely! It was certainly no love-fest between long-lost brown-skinned soul-mates, as it is often portrayed today by the delusional Aztlaners today!
By 1821, Mexico City was strong enough to overthrow even more decrepit and ineffectual Spanish rule. However, the distant provinces of the current US Southwest were far beyond the reach of the authority of independent but strife-torn Mexico. These distant northern provinces received neither military protection nor needed levels of trade from the south.
For example, Comanches ran rampant in the 1830s in this new power vacuum, burning scores of major ranches that had been around for hundreds of years and massacring their inhabitants. Mexico City could neither defend nor keep the allegiance of its nominal citizens in these regions. Nor could it provide needed levels of trade to sustain the prior Spanish-era standard of living.
Meanwhile, a growing America was making great inroads into the Southwest, via ships into California, and via gigantic wagon trains of trade goods over the Santa Fe Trail from St. Louis. The standard of living of the SPANISH in these states subsequently increased enormously, which is why they did not support Mexico City in the 1846-48 war. In fact, the Spanish-speaking inhabitants of the Southwest NEVER considered themselves "Mexicans" at all, ever. They went, in their own eyes, from SPANISH directly to AMERICAN.
So how long did Mexico City have even nominal control over the Southwest? For only 25 years, during which they had no effective control, and the area slipped backwards until the arrival of the Americans. Since then, how long has the area been under firm American control? For 150 continuous years.
And now, we are supposed to let any Mexican from Chiapas, Michoacan or Yucatan march into the American Southwest, and make some "historical claim" of a right to live there? From where does this absurd idea spring?
At what point in history did Indians from Zacatecas or Durango stake a claim on the American Southwest? Neither they nor their ancestors ever lived for one single day in the American Southwest. The Spanish living in the Southwest in 1846 stayed there, and became Americans by the treaty of Guadalupe Hidalgo.
No current inhabitants of Mexico have ANY claim on even one single inch of the Southwest!
They are criminal invaders and colonizers, pure and simple.
Why don't they march for return of the grandparents tumbleweeds 
instead of trying got rob infrastructure that was built by the rest of the world, that doesn't belong to them ?