I asked before this question on MSE but I was not able to work out the details on my own.

Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map $$ T_pM \ni v \mapsto \exp_p(tv)\in M $$ where $t \in (0, \text{inj}(M))$ is a fixed parameter and $\text{inj}(M)$ is the (positive) injectivity radius of $M$.

Is is true that this map in Lipschitz uniformly in $p$? More precisely, is it possibile to prove that there exist $\delta>0$ and $C>0$ s.t. $$d(\exp_p(tv_1), \exp_p(tv_2)) \le Ct \|v_1-v_2\|_p $$

for every $v_1, v_2 \in T_pM$, $p \in M$ and $t \in (0, \delta)$? Clearly with $d$ I mean the Riemannian distance on $M$ and with $\| \cdot \|_p$ the norm in $T_pM$ induced by the Riemannian metric.