They were absorbed by either Pakistan or India. For the most part this was done with no great fuss. The princes received stipends and where allowed to keep honorifics. Initially many of them were regularized and brought into the government. Large princely states continued to be ruled by their princes. Smaller states were consolidated and one of the princes made Rajpramukh (prince-governor). These only lasted until the states were formed in 1956. And over time both the stipends and the honorifics were reduced. By 1972 both India and Pakistan had fully de-recognized the panoply of royal families.

There were a handful of states that were large enough that they attempted to retain full sovereignty, notably Hyderabad which was forcibly annexed to India, Kalat which was forcibly annexed to Pakistan, and Kashmir & Jammu which was complicated and eventually forcibly divided between the two states. But like I said for the most part it was done peacefully.