For the Grassmannian embeddings defined by the spaces \(H^0(X,L^p\otimes E)\), where \(L\) is a positive line bundle and \(E\) is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and the theory of meromorphic transforms developed by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. In the determinant case, we also present an alternative method based on the Wishart distribution and derive variance estimates.