Let $(M,g)$ be a complete Riemannian manifold with sectional curvatures constrained within $[\kappa_{\min},\kappa_{\max}]$. Suppose $x,y\in M$ are two points in $M$ and $v_x\in T_{x}M$ is a tangent vector. Define $\Gamma_{x\to y}:T_{x}M\to T_{y}M$ as the parallel transport map along the shortest geodesic connecting $x$ and $y$. Let $v_{y}=\Gamma_{x\to y}v_x\in T_{y}M$. Our question is whether the following inequality holds:
$$
d({\rm Exp}(x,t v_x),{\rm Exp}(y,tv_y))\leq c_1e^{c_2t}\cdot d(x,y),\quad \forall t\geq 0, \forall x,y\in M,
$$
where $c_1,c_2$ are positive constants independent of $t,x,y$ and $d(\cdot,\cdot)$ is the geodesic distance. 

If needed, the statement can be adjusted. The primary aim is to determine if ${\rm Exp}(x,t v_x)$ converges "uniformly" to ${\rm Exp}(y,tv_y)$ as $x$ tends to $y$. To facilitate this analysis, one may additionally assume that $d(x,y) \leq \epsilon$ for some small $\epsilon > 0$ when required.