The problem of performing inference on the parameters of a functional mixed effect model for multivariate functional data is addressed, motivated by the analysis of 3D acceleration curves of trotting horses. Inference is performed in a local perspective, i.e., defining an adjusted p-value function on the same domain as the data. Such adjusted p-value functions can be thresholded at level α to select the regions of the domain and the coordinates of functional data presenting statistically significant effects. The probability of wrongly selecting as significant a region of the domain, and/or a coordinate of functional data where the null hypothesis is true, is always lower than the pre-specified level α due to the interval-wise control of the family-wise error rate. The procedure is based on nonparametric permutation tests, based on different permutation strategies. It is shown by simulations that all strategies proposed gain in power by taking random effects into account in permutations. Finally, the procedure is applied to the acceleration curves of trotting horses for testing differences between different levels of induced lameness. The method can clearly identify group differences.