It is well-known that there is a duality between affine Demazure modules and the spaces of sections of line bundles on Schubert varieties in affine Grassmannians. This should be regarded as a local theory. In this talk, I will explain an algebraic theory of global Demazure modules of twisted current algebras. Moreover, these modules are dual to the spaces of sections of line bundles on Beilinson-Drinfeld Schubert varieties of certain parahoric groups schemes, where the factorizations of global Demazure modules are compatible with the factorizations of line bundles. This generalizes the work of Dumanski-Feigin-Finkelberg in the untwisted setting. In order to establish this duality in the twisted case, following the works of Zhu, we prove the flatness of BD Schubert varieties, and establish factorizable and equivariant structures on the rigidified line bundles over BD Grassmannians of these parahoric group schemes. This work is joint with Huanhuan Yu.