WARSAW—Tens of thousands of Poles marched across downtown Warsaw on Saturday, in an independence-day procession organized by a nationalist youth movement that seeks an ethnically pure Poland with fewer Jews or Muslims.

The largely young crowd shot off roman candles and many chanted “fatherland,” carrying banners that read “White Europe,” “Europe Will Be White” and “Clean Blood.” Some of the marchers flew in from Hungary, Slovakia and Spain and waved flags and symbols that those countries used during their wartime collaboration...