The number of RNAPs in bacterial cells (e.g., E. coli) have been shown to be smaller than the number of sigma factors. Consequently, if a certain sigma factor is overexpressed, not only will increase the expression levels of genes whose promoters have preference for that sigma factor, but it will also reduce the probability that genes with promoters with preference for other sigma factors.
Meanwhile, transcription initiation has two major rate limiting steps: the closed and the open complex formation. However, only the dynamics of the first step depends on the concentration of sigma factors. Interestingly, the fastest is the closed complex formation relative to the open complex formation, the less responsive is a promoter to changes in sigma factors’ concentration (see  for a model and empirical data of this phenomenon).