Dose-response experiments are widely used in scientific research and nonmonotone dose-response curves are commonly observed. A class of nonparametric tests for dose-response curves subject to downturns at high doses is examined via its asymptotic properties. This class includes the well-known Jonckheere-Terpstra test as a limiting case. The analysis indicates that the Jonckheere-Terpstra test lacks robustness to a downturn in the dose-response, and that other members of the class provide superior overall performance. A result concerning $U$ statistics in locally asymptotically normal families of distributions facilitates the derivation of the asymptotic power function for the class of tests under consideration. This result may also be used to obtain asymptotic power functions for other tests based on $U$ statistics, for instance, the Mann-Whitney-Wilcoxon test. The accuracy of the asymptotic approximation is examined via Monte Carlo simulation and is found to be quite good for moderate sample sizes, suggesting that the approximation might reasonably be used for sample-size determinations.