Tuberculosis is one of the major global diseases in terms of both prevalance and mortality. In recent decades, strains of the disease have evolved that are resistant to several, or all, of the drugs used to treat the disease. Drug resistance is conferred by rare mutations, raising the question of how multiple mutations might have arisen in a single strain. Motivated by this question, we study models of branching processes with mutations which generalize the pioneering work of Luria and Delbruck. We look at the sizes of mutant populations in the limit of mutation rates decreasing to zero, and characterize their limiting distribution. The results show a transition between two regimes depending on the relative growth rate of the mutants: in the slow growth regime, the limiting distribution is Gaussian, while if the mutants reproduce quickly enough, it is heavy-tailed.