My first manufacturing district that I created contained computer, microchip and electronic labs. It is fed by 3 mine districts. Uranium, copper and iron. The copper and iron trucks run right through and cross the map to my lab area. The lab area is NOT a district. Maybe it needs to be?
Districts only determine the relationship between supplier and consumer. Cars still take the shortest path to their target. If you want cars to not drive through a district, you need to give them a highway or superhighway around it.