Hey Dan, I hope I can explain this right because it is really a pain like you said.
First thing you have to understand, and perhaps you already do, is that while the name of the banks might be the same, they're are actually separate entities. Let me explain.
Back home, all Bank of America are branch office the Bank of America Corp. so that you can walk into any BofA branch and they would be able to pull up your account and service it as you request.
Here in China however, that's not the case. The Bank of China here in Kunming let's say, only operates in Kunming, they have nothing to do with BoC in Beijing, they can not access any accounts from any where other than accounts opened at branch offices here in Kunming and vise versa. You can't go to Beijing and expect to have them service your account if it's opened here in Kunming. It's like walking into a WellsFargo and asking them to tell you how much money you have in your Chase Manhattan account.
So how do you access your money when you are away from Kunming, you'll need your ATM card. Just simply stick your ATM into any machine and withdraw it the normal way. You'll be subject to the daily limit of course and then there would be the ATM fees that's expected when you are not using your home bank's ATM.
So, in short, as far as I can tell, there isn't a "national" bank here in China. Any account you open here in Kunming you can pretty much assume that it is only accessible here in Kunming.
On a side note, there might be an exception. If you bank with an out of town bank, the situation might be different. For example, Pudong Bank is a Shanghai based bank, however, they do have branches here in Kunming. If you open your account here, you might be able to access it in Shanghai, wouldn't know about a Pudong Bank in Beijing though. It might be worth asking.
Hope this helps.