We develop extensions of the Luria-Delbrück model that explicitly consider non-exponential growth of normal cells and a birth-death process with mean exponential or Gompertz growth of mutants. Death of mutant cells can be important in clones arising during cancer progression. The use of a birth-death process for growth of mutant cells, as opposed to a pure birth process as in previous work on the Luria-Delbrück model, leads to a large increase in the extra Poisson variation in the size of the mutant cell populations, which needs to be addressed in statistical analyses. We also discuss connections with previous work on carcinogenesis models.