In longitudinal data analysis, there is great interest in assessing the impact of predictors on the time-varying trajectory in a response variable. In such settings, an important issue is to account for heterogeneity in the shape of the trajectory among subjects, while allowing the impact of the predictors to vary across subjects. We propose a flexible semiparametric Bayes approach for addressing this issue relying on a local partition process prior, which allows flexible local borrowing of information across subjects. Local hypothesis testing and credible bands are developed for the identification of time windows across which a predictor has a significant impact, while adjusting for multiple comparisons. Posterior computation proceeds via an efficient MCMC algorithm using the exact block Gibbs sampler. The methods are assessed using simulation studies and applied to a yeast cell-cycle gene expression data set.