Suppose $\gamma$ is a curve on a smooth differentiable manifold M, with tangent vector field $\xi^a$. Let $\alpha^a$ denote a complete vector field on M. From the integral curves of $\alpha^a$, we can define a one-parameter group of diffeomorphisms $\Gamma_s: M \to M$ defined such that $\Gamma_s(p) = \Gamma(s, p)$, where the latter is simply the point that is $s$ units along the integral curve of $\alpha^a$ that starts at $p$.
Now apply the diffeomorphism to $\gamma$, so $\gamma \to \gamma' = \Gamma_s(\gamma)$. How does de tangent vector field $\xi^a$ transform? Let $\xi'^a$ denote the tangent vector field to $\gamma'$, then how is this related to the original tangent vector field?
Specifically, is there always some $s \in \mathbb{R}$ such that $\xi'^a = \xi^a + \alpha^a$? If not, when does such an $s$ exist?