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Despite the claims from certain producers of power banks that their products can handle it (pass-through charging), they should not be charged while they are under a load.
The reason lithium-ion battery banks should not be charged while power is simultaneously being drawn from them is because it violates the electrochemisty of how batteries function.
In a lithium-ion rechargeable battery both the positive electrode (cathode) and the negative electrode (anode) bind lithium ions from lithium oxide in the electrolyte that migrate back and forth between the cathode and anode during the charging and discharging process. When the battery is discharging, the lithium ions moving back to the cathode release energy (electrons) in the process and this is how the battery powers your device. When the battery is being charged, the lithium ions migrate back through the electrolyte in the other direction towards the anode.
So, if you charge the battery while it is simultaneously discharging, then you are trying to force the lithium ions to move in opposite directions through the electrolyte at the same time which will cause a lot of stress to the cells and will eventually lead to lithium metal plating which is the dangerous process than can ultimately cause a thermal runaway (chemical combustion), but for the most part will simply shorten the life of the battery dramatically and reduce its capacity.
@Dashmellow, I am trying to reconcile your statements above with my personal experiences. Since a Lithium battery is a Lithium battery is a Lithium battery..., I tried to apply your statements to my personal experience with my Lithium battery powered laptop. The laptop's original Lithium battery lasted 5 years (2 years ago I replaced it) even as I repeatedly and used the laptop while the AC cable was simultaneously connected to the laptop and visibly charging the laptop's battery. In addition, for 7 years now, when I leave for work, the laptop stays home connected to the AC outlet on the wall and with its Lithium battery re-recharging. I haven't experienced any negative consequences from these practices. Regards.