Finiteness of R-equivalence groups of some adjoint classical groups of type

Let F be a field of characteristic different from 2. We construct families of adjoint groups G of type defined over F (but not over k) such that G(F)/R is finite for various fields F which are finitely generated over their prime subfield. We also construct families of examples of such groups G for which G(F)/R≃Z/2Z when F=k(t), and k is (almost) arbitrary. This gives the first examples of adjoint groups G which are not quasi-split nor defined over a global field, such that G(F)/R is a non-trivial finite group.