  Local Fr\'echet regression is a nonparametric regression method for metric
space valued responses and Euclidean predictors, which can be utilized to
obtain estimates of smooth trajectories taking values in general metric spaces
from noisy metric space valued random objects. We derive uniform rates of
convergence, which so far have eluded theoretical analysis of this method, for
both fixed and random target trajectories, where we utilize tools from
empirical processes. These results are shown to be widely applicable in metric
space valued data analysis. In addition to simulations, we provide two
pertinent examples where these results are important: The consistent estimation
of the location of properly defined extrema in metric space valued
trajectories, which we illustrate with the problem of locating the age of
minimum brain connectivity as obtained from fMRI data; Time warping for metric
space valued trajectories, illustrated with yearly age-at-death distributions
for different countries.
