North of the Rio Grande there's a long tradition of religious leaders resisting the imperial aspirations of their Euro-American neighbors, whether by influencing military policies (such as with Neolin, Tenskwatawa, Hillis Hadjo, and Neapope) or through more peaceful means of resisting cultural assimilation (Handsome Lake and Wovoka, for example). Back when the Spain claimed lands that are now part of the United States, they also contended with religious leaders like Popé and Toypurina.
Who were their counterparts south-of-the-border and beyond?
There aren't really many, no. And there are a number of reasons for this, but it mainly has to do with the different role that religion played in the imperial strategies of Spain v. the United States. Religious conversion was a very important part of the Spanish imperial strategy. Serious conversion efforts began immediately after the conquest, and were refocused during a movement known as the "Extirpation of Idolatry" which began in the late 16th century. Indigenous beliefs, where they existed, were stamped out as quickly as possible. Under the new Catholic religion, indigenous people were denied access to positions of clerical authority - specifically so that this could not happen.
The other point to note is that while in the beginning Catholicism was met with resistance by native peoples, later in the colonial period the Catholic church became a kind of counterweight to the abuses of secular authorities. Many catholic priests (notably Bartolomé de las Casas, but also others) became advocates for indigenous rights. This was especially true in Mexico, where the creation of syncretic Catholic traditions like the Virgen de Guadalupe helped subject people see the Church as a positive institution.
Now that said, there were a few movements of indigenous resistance with religious overtones in the Andes. Spain's ability to exert direct control over the Andes was rather limited when compared to Mexico and Central America, and indigenous religious traditions held out much longer. A prime example of this was the Taqui Onkoy (1564), a millenarian movement lead by the kurakas (native nobility) which sought to reject all aspects of Spanish culture in order to purify the people and pave the way for the restoration of the Inca Empire. Followers of the Taqui Onkoy would break out into fits of shaking during trances -- a practice oddly reminiscent of the "Ghost Dance" movement among the indigenous people of the United States. There were a few other rebellions that weren't quite religious, but sought to tie into indigenous cultural elements as a means of resisting Colonial rule. Tupac Amaru II, an indigenous nobleman from Cuzco, proclaimed himself the next Inca emperor in 1780 and lead a large scale revolt against the colonial authority. All of these movements, however, were quickly put down by the Spanish colonial authorities. Towards the end of the 18th century, the Spanish banned all symbols of indigenous nobility in their colonies in order to prevent this kind of thing from happening again.