I wonder if the following holds in an arbitrary Riemannian manifold $M$:

assume $x\in M$, $h\in T_x M$, do we have for $u\in T_x M$ exponentiable (if necessary of small enough norm) that:

$$\lim_{t\to 0} \frac{d(\exp_{\exp_x u}t\tau(h), \exp_x (u+th))}{t}=0$$ where $\tau(h)$ is the parallel transport of $h$ along $[0,1]\rightarrow M, t\mapsto \exp_x tu$.

If it doesn't hold in general, then which geometrical conditions are sufficient for it to hold, and do we have at least in general that for any $x\in M$, $h\in T_x M$: $$\lim_{t\to 0, u\to 0} \frac{d(\exp_{\exp_x u}t\tau(h), \exp_x (u+th))}{t}=0$$