The question of how natural selection affects asexual mutation rates has been considered since the 1930s, yet our understanding continues to deepen. The distribution of mutation rates observed in natural bacteria remains unexplained. It is well known that environmental constancy can favor minimal mutation rates. In contrast, environmental fluctuation (e.g., at period T) can create indirect selective pressure for stronger mutators: genes modifying mutation rate may "hitchhike" to greater frequency along with environmentally favored mutations they produce. This article extends a well-known model of Leigh to consider fitness genes with multiple mutable sites (call the number of such sites alpha). The phenotypic effect of such a gene is enabled if all sites are in a certain state and disabled otherwise. The effects of multiple deleterious loci are also included (call the number of such loci gamma). The analysis calculates the indirect selective effects experienced by a gene inducing various mutation rates for given values of alpha, gamma, and T. Finite-population simulations validate these results and let us examine the interaction of drift with hitchhiking selection. We close by commenting on the importance of other factors, such as spatiotemporal variation, and on the origin of variation in mutation rates.