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?