Kimberling defined a self-generating set S of integers as follows. Assume 1 is a member of S and if x is in S then 2x and 4x-1 are also in S. We study similar self-generating sets of integers whose generating functions come from a class of affine functions for which the coefficients of x are powers of a fixed base. We prove that for any positive integer m the resulting sequence, reduced modulo m, is the image of an infinite word that is the fixed point of a morphism over a finite alphabet. We also prove that the resulting characteristic sequence of S is the image of the fixed point of a morphism of constant length, and is therefore automatic. We then give several examples of self-generating sets whose expansions in a certain base are characterized by sequences of integers with missing blocks of digits. This expands upon earlier work by Allouche, Shallit, and Skordev. Finally, we give another possible generalization of the original set of Kimberling.