OK, one source of confusion is that cell phone 'chargers' are not chargers - they're just power supplies. The charging circuit is built onto the phone (or istick, etc), and simply needs the correct voltage supply with enough current capability. Power supplies output a fixed voltage and have a maximum current rating.
Contrast that to an actual charger, which has to regulate current so as not to overload the battery. During the constant current phase, it tries to output a fixed current. It does that by varying the voltage, setting it a little above the charge in the battery.
So...
A charger that puts out less current than the maximum the battery can take, will usually just take longer to charge it. No problem.
A power supply that can't put out as much current as the device (charger) wants, will be stressed. It will probably get warm/hot, have reduced voltage output, and possibly melt/burn. Or shut down completely (if it's a good one, but few are). The battery being charged may or may not reach full charge (depending on lots of factors), but is unlikely to be damaged.