Hey Guys! Soooo I studied this a lot. And actually ended up applying to some fellowships but due to some personal/family issues I wasn't able to attend.
Your main options are:
1. Doris Duke - There are 6 schools that host Doris Dukes, you can apply to all 6 if you choose. Only UCSF requires you to find a mentor previous to applying and actually write a research proposal. I applied to the other 5 (Harvard, Yale, Duke, UNC, and Minnesota) and received 3 interviews from them. My global health experience isn't crazy amazing but I did run an independent project after my M1 year. Applications are due beginning of January.
2. Fogarty Fellowship - These are competitive because you are also competing against residents and fellows, not just medical students like with DD. In this case there are ~5 consortium of schools (They all have funny names and some actually make sense to be consortium together, like the UC schools, while others seem random). For the Fogarty Fellowship, you can only apply to 1 consortium and only 2 or 3 actually allow people from outside of the consortium to apply. You'd have to do your own research on this. This requires you to write a full research proposal and submit it. Much more time consuming. I did apply for and get an interview for this as well. Applications are due anywhere from beginning of December to end of January depending on consortium.
3. Fulbright-Fogarty Fellowship - This does not exist this year because it did not get funding, but it will for next year! You apply through Fulbright for this and it acts much like a Fulbright Fellowship, in that you must contact mentors you are interested in working with and get there approval/help in the development of the project. This is similar to the regular Fogarty in this regard. These are also not only for medical students but I believe MPH and public health PhD students can apply for these as well. They are only offered in certain countries so you'd have to go on the Fulbright website to determine which countries are offering them for the coming year. Applications open in September and are due at the end of the year I believe.
From all the research I've done these are the only 3 options open to medical students located anywhere. If you have any questions about it please feel free to ask them here.