I have not gone into the same detail as Visitor here, just done a quick survey of the power banks that I have.
There are 5 in total, in my arsenal, ranging from expensive to cheapo.
Mophie Powerstation XXL (switched):
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iHave (x2) (switched):
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Fuel (unswitched):
Staples 'special' (unswitched):
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For the cursory test, I powered a Chinese Arduino Nano V3, which simply samples A0 to A7, pulled to GND, and outputs the values to the serial port.......:
......using a Chinese knock-off of the Keweisi USB tetster:
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The simple test comprises the current draw from both the Keweisi tester and the Arduino Nano.
The (switched) Mophie switches off on all 3 of the available outlets - 1A, 2.1A and pass-through.
Both of the (switched) iHave units switch off.
The (unswitched) Staples 'special' switches off.
The (unswitched) Fuel powerbank/flashlight combo does not switch off.
The Keweisi tester, if it is to be believed, shows "0.00A", so less than 10mA.
Unless someone REALLY needs to see what the Nano is actually drawing, I will leave it at that and say that in my experience, with the powerbanks I have, it does not appear that all of one type (switched/unswitched) behave the same.