Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p > 0. Using Poincare series, we globalize supercuspidal representations of H-F in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands-Shahidi gamma-factors and the local Langlands correspondence for classical groups.