GÖRLITZ, Germany—As evening fell on a recent Monday in this eastern German city of Gothic spires and Renaissance museums, hundreds of protesters began to gather, just as they have nearly every Monday for at least two years. They carried banners calling for Germany to leave the European Union and cheered speakers who demanded that the nearby border with Poland be shut. They are angry about migrants settling in their communities and inflation squeezing their pensions. They oppose arming Ukraine and say Russian President Vladimir Putin has been unfairly maligned. This is Germany’s rising far right, a movement gaining steam, particularly...