Understanding the 1-parameter subgroup action

Question

Let $G$ be a Lie group and $H$ a closed subgroup. Suppose that $G/H$ has a maximal foliation $\mathcal F$. Let $X\in \mathfrak g$ and let $g^X_t$ be the 1-parameter subgroup of $X$. Let $L$ be the leaf through the point $eH$ and $l\in L$. How to show that if $g^X_{t_0}\cdot l\in L$ then $g^X_{t}\cdot l\in L$ for all $t\in \mathbb R$?