We consider the estimation of a dynamic distribution regression panel data model with heterogeneous coefficients across units. The objects of primary interest are specific functionals of these coefficients. These include predicted actual and stationary distributions of the outcome variable and quantile treatment effects. Coefficients and their functionals are estimated via fixed effect methods. We investigate how these functionals vary in response to changes in initial conditions or covariate values. We also identify a uniformity issue related to the robustness of inference to the unknown degree of heterogeneity, and propose a cross-sectional bootstrap method for uniformly valid inference on function-valued objects. Employing PSID annual labor income data we illustrate some important empirical issues we can address. We first quantify the impact of a negative labor income shock on the distribution of future labor income. We also 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, and how this leads to substantial variation in individuals' incidences to be trapped in poverty. We also provide simulation evidence confirming that our procedures work well.