It very much depended on their attitudes towards the conquerors. If they were good chaps and collaborated with the new overlords, they would have no problems in keeping their local powers, as they would be extremely useful allies for the control of the territory. I can recall several different examples from different points in history.

When the muslim armies under Tariq ibn Ziyad and Musa ibn Nusayr conquered Hispania, the feat was surprisingly easy due to the fact they had the co-operation of the local aristocracies who were against king Roderic of the Goths. The muslim army had surprisingly small numbers, but king Roderic was not exactly popular, being an usurper and all. So, when the armies came, and after the king of Hispania was defeated and killed in the battle of Wadi-Laqq (also known as battle of Guadalete), a few smart aristocrats decided to swear allegiance to the new bosses, convert to Islam, and keep their posessions. The most notorious cases are Theodomirus of Orihuela and Count Cassius.

Theodomirus was a goth aristocrat who controlled the territory in the Sout-East of the Iberian Peninsula, more or less everything between Mula and Valencia. He signed a pact with the invading muslim armies, under which he would pay a yearly tribute to the new overlords, and convert to Islam, and in exchange he would keep his position as governor of those lands over which he already ruled. This pact spared the region of the unnecessary violence that would have come from a war.

Count Cassius was a Hispano-Roman noble who controlled the middle valley of the river Ebro, which was quite an important region. With no particular loyalty towards Roderic, Witiza, or any other gothic king or usurper, he did the same as Theodomirus: peacefully surrendered and converted, in exchange for keeping his posessions and position. His descendants became of paramount importance in the coming centuries, but not under the Cassius name, as they converted: they became the Banu Qasi.

In the last stage of the Reconquista, we have things going the other way around: muslim rulers converting to christianity and keeping power and privileges. The best example would be Sidi Yahya al Nayar (Cidi Hiaya in the Spanish chroniclles), a cousin of Boabdil, the last emir of Granada. Sidi Yahya became a christian, and defected to the side of the Catholic Monarchs. He was rewarded with the command of part of the armies, a title of nobility, and plenty of wealth. His descendants became the marquesses of Campotéjar.

With the Spanish conquest of the Americas, pretty much the same happened: the nobles and tribes who collaborated with the Spaniards received titles, wealth, honours, and privileges. That's why there are so many indigenous names in the Nobiliario de Indias. A few descendants of those indigenous conquistadors became prominent in their own right: Fernando de Alva Ixtlilxóchitl was an aristocrat and a great chronicler, and so was Inca Garcilaso de la Vega. The descendants of Moctezuma received titles and lands as well.

All of this was very pragmatic in nature. The Spanish generals and officers would marry into the local indigenous aristocracy, as that would legitimise their rule over their newly acquired fiefdoms. That way, their children would have double legitimacy: from their Spanish parents they would be the heirs of the conquistadors sanctioned by the Spanish Crown, and from their mothers they would be related to the old Incan or Aztec nobility, setting a continuity with the previous rulers. This attitude was very much promoted by the Spanish Crown. Francisco Pizarro himself married one of Atahualpa's sister named Quispe Sisa, for example.

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Edit: Thanks for the gold, u/Bluespace1234 !