In the divergence case of Khintchine's theorem, Schmidt obtained an asymptotic formula for the number of rational approximations of bounded height to almost every real number. Using exponential mixing in the space of lattices and tools from geometry of numbers, we prove versions of this theorem for intrinsic Diophantine approximation on quadrics, Grassmann varieties, and other examples of flag varieties. Motivated by this proof, we then generalize the Siegel transform to highest weight representations of semisimple algebraic groups and briefly discuss its role in the above problem. Each such Siegel transform is a finite sum of incomplete Eisenstein series, and the Mellin transform provides a passage to the theory of Eisenstein series.