Well, Pentos has a large agricultural hinterland, with the Velvet Hills and the Flatlands, Braavos has the Braavosian Coastland, Norvos has its hill country, Qohor its forests, Volantis its riverrine empire.

But speaking specifically of the Three Daughters, it’s my belief that the Disputed Lands aren’t blasted wasteland as we are told. They don’t look barren on the maps, and realistically, the populations of Lys, Myr, and Tyrosh couldn’t feed themselves with imports alone. So I think that basically each grab a chunk of the Disputed Lands and get the bulk of their foodstuffs from there, and that the fighting is over an attempt to corner the market and dictate terms to their rivals. 

As for how those hinterlands are administered, beyond the idea that the Archon of Tyrosh and the Council of Magisters for Lys and Myr assert overall political authority, I’m not sure. Given the commercial nature of the Free Cities, my guess is that the magisters divvied up the land amongst themselves and then farm it in vast latifundias.