Maps, particularly modern maps of historical polities, offer a false sense of definition. Even with a well-defined entity like the Roman Empire, historical borders were not like modern ones, drawn with GPS and manned by people with assault rifles and no sense of humor. It was rarely the case where you could hop back and forth over a dividing line. This is because, practically speaking, a polity's control over a region was analog, not digital, decaying the farther it got from that polity's base of power.

As for who lived in those "uncontrolled" spaces, the answer varies with time and geography, but the important point is to not think of map borders as a sudden termination of habitation or civilization. They merely represent approximate extents of control.

I'd have to see the maps, because the reasons can differ. Obviously there are deserts, which was likely the case of the Mali empire, as it was located in the Sahara desert. In the Mongol empire there were also deserts, steppes and high mountains.

Another obvious reason for blank spaces was that the peoples and cities were simply unknown to ancient geographers. For that reason often only the coast was shown, with the interior remaining unmapped. Often there were entire empty continents that didn't even exist, as in the case of Terra Australis.