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This isn't trivial in Tableau due to the way Tableau works. Tableau uses the arrangement of the blue (discrete) pills on Rows and Columns to draw headers, then it draws an axis for each green (continuous) pill. Tableau draws maps by using continuous axes for latitude and longitude, then draws a mark at each "cell" determined by the level of detail in the view, which is made up of all the blue (discrete) pills in the view plus any green pills that are dimensions. In the case of a standard map with Country on the Level of Detail, there's only one mark per country.
What you're asking Tableau to do is draw a time series at each point in the map, and that's not quite how Tableau is built at present. There's a feature request to do something on a related note here: http://community.tableau.com/ideas/1181.
However, there is a workaround involving table calculations that could work for you, here's something Andy Cotgreave put together re: sparkbars in tooltips, you can also put them in mark labels: http://interworks.co.uk/andy-cotgreave/sparkbar-tooltips/. The link notes the major caveat, which is that it requires a font to be installed on every client machine.
Thanks Jonathan! Sad to know that you can't do this without a serious workaround. Oh well, maybe in Tableau 8
Sorry to disappoint, it's not in Tableau 8 either. Maybe Tableau 9?
One thing Tableau 8 will allow for is floating elements on a dashboard. So there is a possibility you could use that to accomplish your goal.
1. There would be a worksheet for each country.
2. The map would need to be fixed (or you could use your own map as a background image) -- but there is no way to disable the ability for the end user to pan or zoom the map (and the floating elements are not tied to a point on the map, but rather coordinate on the screen).
3. It could be fairly dynamic in terms filtering the various countries -- you'd want to hide the title for each floating view.
4. Positioning might be difficult to avoid overlap, especially if you had a lot of countries or were working with a very tight view.
5. The floating elements have no option for transparency, so they will obscure anything under them (you're fine if you can place them on an ocean like the above image, but it probably won't look great otherwise).
I'd love to hear from you once you start using version 8 to see if this approach might work in your circumstance.