I think it truly depends on a variety of things, including your field/interests/country on how you go out setting up the affiliation.
For me, I definitely allowed what seemed to be the best affiliation to drive my location. Before I even started thinking about the application, I knew that I was interested in a specific university in a specific country. From that point, I sent emails to the major researchers in my field at the university. I sent them my C.V., information about the Fulbright Fellowship, and other information about my interests. I only sent out around 4 emails at first and received a response. She was quite enthusiastic and gladly helped design a project with me. I came up with an idea of a project that fit within her interests and her expertise. She provided a lot of feedback on the design and wrote a great letter of affiliation.
The only thing is that it literally took months and months for this to finalize I started sending out emails at the end of May/beginning of June and only finalized everything with my potential affiliate at the beginning/middle of September.
