We study certain relevant boundary perturbations of Liouville theory and discuss implications of our results for the brane dynamics in noncritical string theories. Our results include (i) There exist monodromies in the parameter μ_B of the Neumann-type boundary condition that can create an admixture represented by the Dirichlet type boundary condition, for example. (ii) Certain renormalization group flows can be studied perturbatively, which allows us to determine the results of the corresponding brane decays. (iii) There exists a simple renormalization group flow that can be calculated exactly. In all the cases that we have studied the renormalization group flow acts like a covering transformation for the monodromies mentioned under (i).