We consider local-global principles for homogenous spaces under rational linear algebraic groups defined over function fields of curves over a complete discretely valued field. In previous work we proved such a principle for connected groups in the context of patching, and obtained applications to u- invariants of fields (concerning quadratic forms) and to the period-index problem (concerning Brauer groups). Here we consider variant principles in terms of completions at discrete valuations and also in the case of disconnected groups, where the reduction graph plays a key role. (This is joint work with Julia Hartmann and Daniel Krashen.)