Assuming control over districts is a complex, most time-consuming, as well as the most difficult from all the activities present in the game. The hole mechanism of recapturing consecutive sections of Seattle resembles in many ways that of the game Red Faction: Guerilla, because it implies retrieving the districts formerly in the D.U.P.'s hands. The first step should be destroying the enemy base in your selected district, and an example of this process can be the ending part of the first main mission after arriving in Seattle (Welcome to Seattle). You have to locate and blow up the mobile command center, as well as get rid of all the enemy soldiers in the area.
After taking care of the enemy base, you should start doing additional activities in that district, which was described thoroughly in the "Step 8" section. You don't have to complete 100 percent of the activities, but you shouldn't skip them, because more than once you will receive rewards after completing them.
Activities mentioned above (destroying the base, doing additional tasks) have to be done until the percentage showing the influence of the D.U.P. forces falls below 30% - the current progress can be checked from the world map. Afterwards, you have to get to a place marked by a phone on the map (as can be seen on the above screenshot) and after making a call get ready for an encounter for the control over the district (District Showdown).
All the fights connected with the retrieval of the district are quite challenging, because the surrounding area will be literally flooded with the D.U.P. forces. I would recommend to look into the "Step 2" and "Step 3" beforehand. First, you should get rid of the enemy armored forces (tanks, armored vehicles, helicopters) and supernatural-endowed troops - don't bother with the regular enemies before doing it. During the fights, you can retreat a little from the battlefield (to regenerate your smoke supplies, for instance), but pay attention not to get outside the area marked on map, because you will fail your mission.
Eliminating all the enemy forces in the area is equivalent to retrieving the district, and it's awarded in two ways. First, the game allows you to use fast travel system in all retrieved districts. Secondly, you won't run into enemy forces in those areas, so you can explore the area freely.