Many longitudinal imaging studies have been/are being widely conducted to use diffusion tensor imaging (DTI) to better understand white matter maturation in normal controls and diseased subjects. There is an urgent demand for the development of statistical methods for analyzing diffusion properties along major fiber tracts obtained from longitudinal DTI studies. Jointly analyzing fiber-tract diffusion properties and covariates from longitudinal studies raises several major challenges including (i) infinite-dimensional functional response data, (ii) complex spatial-temporal correlation structure, and (iii) complex spatial smoothness. To address these challenges, this article is to develop a longitudinal functional analysis framework (LFAF) to delineate the dynamic changes of diffusion properties along major fiber tracts and their association with a set of covariates of interest (e.g., age and group status) and the structure of the variability of these white matter tract properties in various longitudinal studies. Our LFAF consists of a functional mixed effects model for addressing all three challenges, an efficient method for spatially smoothing varying coefficient functions, an estimation method for estimating the spatial-temporal correlation structure, a test procedure with a global test statistic for testing hypotheses of interest associated with functional response, and a simultaneous confidence band for quantifying the uncertainty in the estimated coefficient functions. Simulated data are used to evaluate the finite sample performance of LFAF and to demonstrate that LFAF significantly outperforms a voxel-wise mixed model method. We apply LFAF to study the spatial-temporal dynamics of white-matter fiber tracts in a clinical study of neurodevelopment.