Only if they or others are in direct harm, first use chemical agents, then water guns, then non-lethal methods and THEN shooting as a last resort.
During soccer riots which usually are a LOT more violent, especially if they are between Ajax Amsterdam
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As it says on this image, we are AJAX and we are the BEST and the other big club of the Netherlands, Feyenoord are our arch nemesis. Well we call them second best but they think we from Ajax are their arch enemies. Hence the fact that Ajax fans cannot visit Feyenoord matches and the other way around.
If the Dutch police can stop riots and disperse rioting crowds with just the help of the ME (mobile unit, officers specially trained in vans hence the world mobile that can travel to where ever they are needed in a city/area), regular police, dog units and horse units, without firing a single shot in 99.99% of the cases. Then the US police should be able to stop looters without shooting people.
If used properly, dog teams/horse teams/trained riot police, can stop most violent mobs without guns. Maybe US police should invest in top of the line water gun vehicles rather than take military surplus IMO. Water guns can stop a lot of protesters as it is not fun to be blasted by a water jet.