There's not reliable evidence linking the Natchez with any Mesoamerican society. There is part of the oral history that does mention coming from the south - the earliest written version from the mid-18th Century even identifies this area as Mexico - but it's not so simple.

Specifically, the this narrative mentions that the Natchez expanded southward first. The writer (Du Pratz, who is our main source of on the Natchez - see either his History of Louisiana or Swanton's Indian Tribes of the Lower Mississippi Valley and Adjacent Coast... which quotes Du Pratz heavily for the Natchez) says he thinks they were talking about Mexico, but we have to remember that at the time Mexico included Texas and based on the description given it seems the Natchez were talking about the coastal Texas, which is not terribly far away from the historic homeland just north of the mouth of the Mississippi. In the narrative, the Natchez attempt to settle this area but are eventually opposed by the Ancients of the Land - great artisans who lived in stone houses as large as whole villages, and who inhabited the area as far as the western sea. This may mean that the Ancients were Pueblo and related peoples from further up the Rio Grande, or they may be a generic amalgamation of all peoples that live south and west of the Natchez. In time, the Great Sun - the leader of the Natchez - realized they couldn't win against the Ancients so he ordered a group to head north again and re-establish a presence on the Mississippi as a fall back position. Once this was set up, the Great Sun pulled his people out of the area and returned to the Mississippi.

However, there is another incident with the Ancients that may indicate relatively recent communication with Mesoamerica. The story goes that at one of the Ancients decided that he should be lord over all the others and warred against his neighbors. Eventually the Warriors of Fire showed up among the Ancients. This is the name the Natchez used for the Spanish, whom they first encountered in the 1540s (De Soto's entrada). The other Ancients allied with the Warriors of Fire and sent an invitation to the Natchez to join them against the would-be lord. The Great Sun refused, expressing concerns that these new allies would turn against them the moment the common enemy was defeated. The story ends with "And this is what we have heard has happened." We can't be certain that the Natchez had contact with Mesoamerica around the time of Cortes' conquest. It's also possible that men who were left behind during de Soto's entrada relayed this information, which was then incorporated into Natchez oral history to be eventually written down by a French history nearly two centuries later.

As for human sacrifice, yes, the Natchez did perform human sacrifices but for a very different purposes than those performed in Mesoamerica. For the Natchez, human sacrifices were only performed during certain funerals. When one of the Sun's died, they would be accompanied by an entourage on a semi-voluntary basis. Servants of the Suns were, let's say, strongly encouraged to join their masters in the next life, but we know of at least one who decided to skip his master's funeral and received no punishment. While Du Pratz lived in Louisiana he was present for the funeral of the Tattooed (sometimes "Stung") Serpent, who was the leader of the Natchez during times of war. He was the second most important man among the Natchez after the Great Sun himself, who was his brother. The Great Sun wanted to join his brother in the next life and had to be dissuaded on multiple occasions. Even after he was convinced that it was in everyone's best interest if he remained in this world to lead the Natchez, he had to be put on a sort of suicide watch to make sure he didn't side step the usual ceremony that went into being a funerary sacrifice and take matters into his own hands. Those that did go to the grave with the Tattooed serpent included his sister, two wives, his second-in-command, his physician, the head of his servants and his wife, and a few others.