I deal with a curve $\gamma$ in a Riemannian manifold $M$, taking values in the domain of a local chart. I suppose that every couple of points $(x,y)$ can be joined with a unique geodesic with minimal length (I denote by $P_{x,y}$ the parallel transport according to this geodesic and the Levi-Civita connection). I define for all $\alpha>0$ the following parallel transport: $$ P_{0,a}^\alpha(\gamma)=P_{\gamma(\nu^\alpha (a)),\gamma(a)}...P_{\gamma(\alpha),\gamma(2\alpha)}P_{\gamma(0),\gamma(\alpha)},$$ where $\nu^\alpha(a)$ is the biggest real $k\alpha$ ($k$ is an integer) less or equal to $a$.
Would it be possible to compare in local coordinates the transports according two parameters $\alpha$ and $\beta$ ? More precisely, I would like an estimate on: $$ \mid P_{0,a}^\alpha(\gamma)u-P_{0,a}^\beta(\gamma)u\mid$$ where $u \in T_{\gamma(0)}M$.