The Schubert classes on $G/P$ are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique $T$-fixed point. The $T$-fixed points on $G/P$ are the images of $T$-fixed points on $G/B$ (since $T$ acts on the fiber, which is a projective variety, hence itself has a $T$-fixed point by Borel's theorem).

Up on $G/B$, the $T$-fixed points are exactly of the form $N_G(T)B/B$, so indexed by the Weyl group $W_G = N_G(T)/T$. Down on $G/P$, they group together by the Weyl group $W_P = N_P(T)/T$, so they're indexed by $W_G/W_P$. Which is exactly what you observed in the $G/P =$ Grassmannian case.

(Actually you asked about compact groups, so $K/L$ where $K$ is compact and $L$ is compact of the same rank, which includes some cases like $S^4 = SO(5)/SO(4)$ that is not of the form $G/P$ for $G$ complex and $P$ parabolic. Then there's still a basis of "Schubert classes", indexed by $W_K/W_L$ similarly.)