We consider estimation of a dynamic distribution regression panel data model with heterogeneous coefficients across units. The objects of interest are functionals of these coefficients including linear projections on unit level covariates. We also consider predicted actual and stationary distributions of the outcome variable. We investigate how changes in initial conditions or covariate values affect these objects. Coefficients and their functionals are estimated via fixed effect methods, which are debiased to deal with the incidental parameter problem. We propose a cross-sectional bootstrap method for uniformly valid inference on function-valued objects. This avoids coefficient re-estimation and is shown to be consistent for a large class of data generating processes. We employ PSID annual labor income data to illustrate various important empirical issues we can address. We first predict the impact of a reduction in income on future income via hypothetical tax policies. Second, we examine the impact on the distribution of labor income from increasing the education level of a chosen group of workers. Finally, we demonstrate the existence of heterogeneity in income mobility, which leads to substantial variation in individuals' incidences to be trapped in poverty. We also provide simulation evidence confirming that our procedures work well..