The cities you are not playing currently are frozen in time. However, if you volunteer your services to another city, that city will be able to use them regardless of the city being active or not. You can absolutely volunteer all of your services to one city, and then change those when you want to work on another city.
The only caveat here is that you have to have a road connection to that city in order to volunteer the vehicles. So you cannot volunteer every vehicle in a region to one city. Not all cities have direct connections. It can be a bit annoying because you may not have a direct connection to a city that is right next door.
Also, when you volunteer your services to another city that you aren't playing, you get money from that city for the services. I've not seen any situation where you pay money for services volunteered however.