I suppose the fact that Augustus is the last censor mentioned (alongside Tiberius Julius Caesar) is a hint that the emperor assumed the role of overseer of public morality and other attributes of the censor's role. What about the census though? Were the duties spread amongst lower administrators or did its nature change completely? When did the practice stop altogether? The list itself is to be found here
There's no great mystery: the censorship was abolished and its duties granted to the princeps. The censorship was already a dying office, and the census was irregular in the first century. Sulla abolished the office and assumed censorial powers, but the office was reestablished in 70. Caesar abolished it again during his dictatorship, assuming the extraordinary office of praefectus morum, and though Augustus technically revived the office in reality the censorship lapsed, and when it was necessary for there to be censors the emperors generally just assumed the power for themselves or appointed prefects. This is no great surprise, as censorial elections had been very irregular by the first century, and moreover the census had become increasingly difficult--in some lustra the censors had even had difficulty completing the ritual lustration, which was the more important of the censors' main duties