  This paper gives a contribution to the study of regularity of Lagrangian
flows on non-smooth spaces with lower Ricci curvature bounds. The main
novelties with respect to the existing literature are the better behaviour with
respect to time and the local nature of the regularity estimates. These are
obtained by sharpening previous results of the first and third authors, in
combination with some tools recently developed by the second author (adapting
to the synthetic framework ideas introduced in [Colding-Naber 12]. The
estimates are suitable for applications to the fine study of $\text{RCD}$
spaces and play a central role in the construction of a parallel transport in
this setting.
