Over number fields, according to work of Sansuc and Borovoi, the Brauer-Manin obstruction is the only obstruction to the local-global principle for homogeneous spaces under connected linear groups with connected stabilizers. In the last years, there has been a growing interest in the local-global principle over other fields coming from geometry. Interesting examples of such fields are function fields of curves over a complete discretely valued field, and Laurent series fields in two variables. In this talk, I will report on a recent work with Giancarlo Lucchini Arteche, in which we study the obstructions to the local-global principle for homogeneous spaces with connected stabilizers over such fields. We will in particular see that, contrary to what happens over number fields, the Brauer-Manin obstruction is not enough to explain the failures to the local-global principle. If time permits, we will then discuss how one can solve this problem by combining the Brauer-Manin obstruction with descent obstructions.