Let $G$ be a real semisimple algebraic group, and $\Gamma$ a lattice in $G$. We will discuss the limiting behavior of translates of a real analytic curve on $G/\Gamma$ by a diagonal flow. We will show that the natural obstructions to non-escape of mass and equidistribution are given by Schubert varieties and partial flag subvarieties. I will explain how the ideas from geometric invariant theory and the work of Kempf are involved in the study of this problem.