HONG KONG—Six months after an estimated one million people marched in an antigovernment rally, large crowds again took to the streets of Hong Kong on Sunday in a renewed sign of demonstrators’ determination to wrest democratic concessions from their leaders.

The mostly peaceful, family-filled rally contrasted with months of growing tension and violence between democracy activists and police leading up to district council elections two weeks ago in which pro-Beijing candidates were routed.

While...