We investigate how the dynamics and outcomes of adaptation by natural selection are affected by environmental stability by simulating adaptive walks in response to an environmental change of fixed magnitude but variable speed. Here we consider monomorphic lineages that adapt by the sequential fixation of beneficial mutations. This is modeled by selecting short RNA sequences for folding stability and secondary structure conservation at increasing temperatures. Using short RNA sequences allows us to describe adaptive outcomes in terms of genotype (sequence) and phenotype (secondary structure) and to follow the dynamics of fitness increase. We find that slower rates of environmental change affect the dynamics of adaptive walks by reducing the fitness effect of fixed beneficial mutations, as well as by increasing the range of time in which the substitutions of largest effect are likely to occur. In addition, adaptation to slower rates of environmental change results in fitter endpoints with fewer possible end phenotypes relative to lineages that adapt to a sudden change. This suggests that care should be taken when experiments using sudden environmental changes are used to make predictions about adaptive responses to gradual change.