eerr not sure what a power puller is, each item has a rating on it, in amps or milliampsYou have to be very careful about the total power draw from the device.
I can charge my tablet from some (not all) multi-devices as long as it is the only item drawing power. I can charge phone, camera, mp3 player all at the same time as they don't pull so much power. Still worth having the multi-charge device as I can charge several low pull items or a single big power puller - rather pack one device than several.
A device will draw power until it is fully charged
The Anker device I mentioned above is smart, it will adjust as required to a max of 40w, [60w if you buy the big one] adjusting each USB as needed
I have burnt out small USB supply devices, they just deliver the rated output and get hot/burn if the demand is too high, unregulated, they are in effect dangerous
I have had a small 12v cig to USB melt and blacken