  The study of the dynamic behavior of cross-sectional ranks over time for
functional data and the ranks of the observed curves at each time point and
their temporal evolution can yield valuable insights into the time dynamics of
functional data. This approach is of interest in various application areas. For
the analysis of the dynamics of ranks, estimation of the cross-sectional ranks
of functional data is a first step. Several statistics of interest for ranked
functional data are proposed. To quantify the evolution of ranks over time, a
model for rank derivatives is introduced, where rank dynamics are decomposed
into two components. One component corresponds to population changes and the
other to individual changes that both affect the rank trajectories of
individuals. The joint asymptotic normality for suitable estimates of these two
components is established. The proposed approaches are illustrated with
simulations and three longitudinal data sets: Growth curves obtained from the
Z\"urich Longitudinal Growth Study, monthly house price data in the US from
1996 to 2015 and Major League Baseball offensive data for the 2017 season.
