The Schubert classes on G/P$G/P$ are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique T$T$-fixed point. The T$T$-fixed points on G/P$G/P$ are the images of T$T$-fixed points on G/B$G/B$ (since T$T$ acts on the fiber, which is a projective variety, hence itself has a T$T$-fixed point by Borel's theorem).
Up on G/B$G/B$, the T$T$-fixed points are exactly of the form N_G(T)B/B$N_G(T)B/B$, so indexed by the Weyl group W_G = N_G(T)/T$W_G = N_G(T)/T$. Down on G/P$G/P$, they group together by the Weyl group W_P = N_P(T)/T$W_P = N_P(T)/T$, so they're indexed by W_G/W_P$W_G/W_P$. Which is exactly what you observed in the G/P =$G/P =$ Grassmannian case.
(Actually you asked about compact groups, so K/L$K/L$ where K$K$ is compact and L$L$ is compact of the same rank, which includes some cases like S^4 = SO(5)/SO(4)$S^4 = SO(5)/SO(4)$ that is not of the form G/P$G/P$ for G$G$ complex and P a$P$ parabolic. Then there's still a basis of "Schubert classes", indexed by W_K/W_L$W_K/W_L$ similarly.)
