The Uto-Aztecan peoples probably migrated from the region northwest Mexico since that is where the greatest variation in its languages is, pointing to that being the area it's been around the longest. The Nahua were not all that different originally from many of the "barbarian" peoples to the north of the Classical civilized area, other than being numerous, and every time there was chaos in central Mexico northern barbarians would come wandering in. If the Nahua went elsewhere another tribe would have probably replaced them like the Otomi (who came in with the Nahua migrations after the fall of the Toltecs) or perhaps a different Uto-Aztecan group (the Huichol most likely). Teotihuacan's ethnic origins are disputed but the Valley of Mexico and surrounding areas seem to have undergone a steady Nahuazation after it fell due to these migrations.
If the Nahua leave their original homeland (likely in San Luis Potosi or the surrounding area) around 500 BC it would take a a few centuries for the main mass to reach the Lower Mississippi, probably around 500 AD-ish at least. They might be able to bring better varieties of corn and beans or more advanced bowmaking technology due to their origins near Mexico. By that time the Mississippi cultures had a few corn varieties already; but they didn't have beans yet and those would be more significant both as food and for their nitrogen-fixing capacities. I have read that the Mississippians might have had soil exhaustion issues which may have caused the period of decline they were in when the Europeans showed up. Stave that off, and perhaps there will be both a more numerous and more organized set of tribes along the rivers when they arrive, though, probably, not one that will stay independent unless they play their cards exceptionally well.