Let $K$ be either a global function field or a function field of an algebraic surface. Johan de Jong formulated the following
principle: a ``rationally simply connected'' $K$-variety admits a rational point if and only if the elementary obstruction
vanishes. In this talk, I will discuss how this principle works for projective homogeneous spaces. In particular, it leads to
a classification-free result towards the quasi-split case of Serre's Conjecture II over $K$.
