Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives, play an important role within social choice theory and the mathematical social sciences at large. Laffond et al. have shown that various tournament solutions satisfy composition-consistency, a structural invariance property based on the similarity of alternatives. We define the decomposition degree of a tournament as a parameter that reflects its decomposability and show that computing any composition-consistent tournament solution is fixed-parameter tractable with respect to the decomposition degree. Furthermore, we experimentally investigate the decomposition degree of two natural distributions of tournaments and its impact on the running time of computing the tournament equilibrium set.