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In this paper we prove that a metric measure space $(X,d,m)$ satisfying thefinite Riemannian curvature-dimension condition ${\sf RCD}(K,N)$ isnon-branching and that tangent cones from the same sequence of rescalings areH\"older continuous along the interior of every geodesic in $X$. Moreprecisely, we show that the geometry of balls of small radius centred in theinterior of any geodesic changes in at most a H\"older continuous way along thegeodesic in pointed Gromov-Hausdorff distance. This improves a result in theRicci limit setting by Colding-Naber where the existence of at least onegeodesic with such properties between any two points is shown. As in the Riccilimit case, this implies that the regular set of an ${\sf RCD}(K,N)$ space has$m$-a.e. constant dimension, a result already established by Bru\`e-Semola, andis $m$-a.e convex. It also implies that the top dimension regular set is weaklyconvex and, therefore, connected. In proving the main theorems, we develop inthe ${\sf RCD}(K,N)$ setting the expected second order interpolation formulafor the distance function along the Regular Lagrangian flow of some vectorfield using its covariant derivative.

H\"older continuity of tangent cones in RCD(K,N) spaces and applications to non-branching