In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via 
$$
\tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi), \  n:=\langle \pi(1) \alpha^\vee \rangle \geq 0 \\ 
e^{-n}_\alpha(\pi), \ n:= \langle \pi(1), \alpha^\vee \rangle < 0  
\end{cases}
$$

For a straight line path $\pi_\lambda(t)=t\lambda$, it is clear that for any $w \in W$ we have $w(\pi_\lambda)=\pi_{w\lambda}$. 

> Is there a clear picture of how $W$ acts on more general paths? For example, what if we take a "dog leg" path 
$$
\pi=\pi_\lambda \ast \pi_\mu,
$$
the concatenation of two straight line paths?