a tiny minority can't but somewhere between 10% or a quarter is enough to pressure the rest to support them. for example to force the majority of restaurants to make sure their food is halal, because otherwise muslims would not eat there. or to port your software to run on macs, etc...

it depends of course on the cost of compliance. halal food is fine, because everyone can eat it. a minority of vegans could not force restaurants to stop serving meat because that would in turn exclude the majority of meat eaters.

I think there is a selection bias here. If it succeedes the minority becomes a majority.

I think a good example are religious movements (obviously i dont want to equate religion with morality, but often they come with certain moral views attached). They start as a minority, and either they die or they take over. Christianity was a minority at one point.