Native Americans prior to European contact had two domesticated draft animals: dogs, and in the South American Andes, llamas. Since your question asks about nomadic cultures, I'll leave discussion of llamas for an Andean expert.
Dogs are the precursor to the horse. As Pekka Hämäläinen states in his overview of Plains Indian Horse Cultures:
...Indians' prior experience with dogs greatly facilitated the incorporation of horses: having turned dogs into their beasts of burden, they were culturally preadapted for the utilization and subjugation of horses as well. The close association between dogs and horses was universal on the northern Plains...
So horses to Native Plains peoples were basically Dog 2.0--bigger, better, faster, more versatile, but a variation on a theme they'd practiced for centuries. This isn't to diminish the enormous changes the horse brought to Native cultures, but to emphasize that the horse was the logical progression of the dog as draft animal, not a completely new paradigm.
Dogs were deliberately bred as pack animals by Native peoples from Canada to Northern Mexico, and had been in use for centuries, and probably millennia, by the time Europeans arrived. Depending on the climate and season, they would pull sledges or travois or be loaded with individual packs. Indigenous dog breeds have mostly vanished through interbreeding with European dogs, but they varied extensively in size, color, and coat length. Some were as large as a northern wolf (80+ pounds) and others a medium size of 20-40 pounds. Breeds further north (Canada and the North-Central US) seem to have been larger on average than those in the Southern Plains (Oklahoma, Texas, New Mexico).
They were bred for strength, endurance, and temperament. European explorers often described Native dogs as barely more domesticated than wolves, frequently attacking people or horses. However, this is probably an outsider's view. Interviews with the Hidatsa of North Dakota from a 1924 ethnography describe dogs familiar to any working farm--they were friendly to their group, bred for trainability and intelligence, and aggressive dogs were put down quickly. Interestingly enough, this same ethnography has the Hidatsa describing the dogs of the Sioux, their enemies, as mean and aggressive. So it's doubtful Indigenous draft dogs were any more bad-tempered than any other working dog.
Further south, the Jumanos and Apaches of the Southern High Plains were described by Spanish explorers as using literally hundreds of draft dogs for trade and travel. One of the earliest accounts in 1601 is of a group of Apaches in Eastern New Mexico or West Texas:
These Indians live in rancherias [communities] which they move from place to place. They pack their huts, which are made of hides, on some small dogs.
The Vaquero Apaches carried on an enormous amount of trade with the Eastern Pueblos, particularly Pecos Pueblo near Santa Fe. They used dogs to haul trade goods, mostly bison hides and meat, probably also pottery and bow-wood from the Caddoans back East, and raw toolstone. The dogs also hauled their own belongings and tents when they travelled.
In the early 1600s, Fray Benavides, a Franciscan monk in New Mexico, describes the Vaquero Apaches travelling down the Rio Grande Valley:
When these Indians go to trade and traffic...the entire rancherias [communities] go...and the tents they carry loaded on pack-trains of dogs, harnessed up with their little pack-saddles, and the dogs are medium-sized...five hundred dogs in one pack train...the people carry their merchandise thus loaded, which they barter for cotton cloth and for other things that they lack [source]
These are just a sampling of sources, but it shows the incredibly widespread use of pack dogs all over the North American Plains. The Hidatsa and Apache, as well as many other Plains nations, were still using their pack dogs well into the nineteenth century. Although horses were superior in many ways, pack dogs were quicker and easier to load and manage, and less expensive and time-consuming to maintain.
Domestic horses were one of the new technologies Europeans brought to the Americas, and Natives immediately saw the advantages. Raising, training and riding horses is complex, and the earliest Native horsemen quickly learned this. Santa Fe was founded by the Spanish in northern New Mexico in 1610, and by the 1650s the Spanish governor was already complaining of attacks by mounted Navajo-Apaches. Within another few decades, the knowledge of horsemanship had spread north into the Midwest Plains and the Pacific coast, and by the early 1700s, you had previously sedentary farming peoples radically changing their lifestyles as they became the "horse cultures" of the Plains and the West.
Sources:
Replicating Dog Travois Travel on the Northern Plains, Norman Henderson
Plains Anthropologist, Vol. 39, No. 148 (May 1994), pp. 145-159
Ethnohistory of the High Plains, James H. Gunnerson & Dolores A. Gunnerson
Plains Anthropologist, Vol. 17, No. 55 (February 1972), pp. 1-10
The Horse and the Dog in Hidatsa culture, Wilson, Gilbert Livingstone
Anthropological papers of the American Museum of Natural History ; v. 15, pt. 2, 1924
The Jumanos: Hunters and Traders of the South Plains, Nancy Parrott Hickerson
University of Texas Press, 1994
The Rise and Fall of Plains Indian Horse Cultures, Pekka Hämäläinen
The Journal of American History, Vol. 90, No. 3 (Dec., 2003), pp. 833-862
I'm going to focus on the indigenous peoples of the United States and Canada for this response, and even then I'm going to be skipping over and generalizing. It's important to remember that many Native American peoples were more sedentary than popular U.S. history teaches, but your question asks about nomadic tribes, so my answer has two parts: talking about dogs as beasts of burden, and acknowledging that nomadic lifestyles do not necessarily need beasts of burden. I'll spend more time on the first part. Let's get into it.
In relatively recent history, North America had no horses, cattle, llamas, camels, or any other species that we would recognize as beasts of burden in modern times. Some of these animals, such as horses and camelids, were present in North America when Native Americans first arrived on the continent, but were driven to extinction by variously undetermined combinations of human hunting and climactic shifts. However, the first Native Americans brought domestic dogs with them onto North America when they crossed from Asia. Many Native American peoples developed these dogs as beasts of burden. This occurred in two main ways. The first is one still common amongst communities in the Arctic today: sled dogs. It seems that sled dogs in the Canadian Arctic became widespread around 1000 AD with the advent of the Thule culture, although earlier groups made intermittent use of dogs for various reasons. I'll quote a relevant article that highlights how important dog sleds and associated technologies were:
The success of the Thule diaspora...was undoubtedly based on numerous technological traits and organizational characteristics, with high mobility and transportation efficiency likely among the most important...Two of these, dog sleds and umiaks, are still unknown from the Paleoeskimo record in North America...recent summaries [emphasize] Thule transportation capabilities, including dog sleds (e.g., Maxwell, 1985; Condon, 1996; McGhee, 1996)...Eastern Arctic Thule sites, at least in all but the southernmost reaches, routinely yield not only numerous bones of dogs, but also sled parts and other associated material remains, including trace buckles and whip shanks.^(1)
Modern dog breeds such as Alaskan Malamutes, Greenland Dogs, and Canadian Eskimo Dogs are descended from pre-European Native American dog breeds. The movement and carrying capacities afforded by sled dogs in the Arctic are significant, and have an ongoing impact on life in the area.
Moving outside of the Arctic, Native Americans still made use of dogs in transportation. Dogs in forested areas adjacent to the Great Plains were fitted with packs to carry small bundles^(2), but an equivalent of the sled was common throughout Native tribes of the Great Plains and mountain West: the travois. Imagine a triangular arrangement of poles, with one tip of the triangle on a dog's back and the other two dragging across the ground. Here are some post-European contact pictures and paintings of Native American travois dogs. Ethnographic records and biological specimen analyses suggest that these dogs could have carried around 45 kilograms (100 lbs) of weight at a time, and were used for more than 3,000 years.^(3)
So dogs were the only beasts of burden in North America prior to European contact. But it's important to remember that dogs or pack animals are by no means necessary for successful nomadic lifestyles. The earliest hints of dog domestication are from around 33,000 years ago, and even those are debated.^(4) Whatever the case, anatomically and behaviorally modern humans existed on Earth long before any relationships with canids. These individuals lived semi-nomadic or completely nomadic lives without animal partners, carrying their entire livelihoods on their own backs. I can't speak to the exact details of those lifestyles, since they varied widely (as do the lives of modern peoples who live nomadic lives without pack animals), but it is clear that they existed (and still exist) and thrived.
^(1) Morey, D., & Aaris-Sørensen, K. (2002). Paleoeskimo Dogs of the Eastern Arctic. Arctic, 55(1), 44-56.
^(2) Henderson, N. (1994). Replicating Dog Travois Travel on the Northern Plains. Plains Anthropologist, 39(148), 145-159.
^(3) Welker, M. H., & Byers, D. A. (2019). The Birch Creek Canids and Dogs as Transport Labor in the Intermountain West. American Antiquity, 84(1), 88-106.
^(4) Wayne, R.K., vonHoldt, B.M. Evolutionary genomics of dog domestication. Mamm Genome 23**,** 3–18 (2012).
