Big consumer brands like Coca-Cola Co. and Delta Air Lines Inc. for years have positioned themselves as forces for promoting what they see as social good—an approach they displayed last summer after the death of George Floyd.

Coca-Cola turned off its Times Square billboard for a day. Delta flew Mr. Floyd’s body to his family in Houston. The Atlanta-based companies were among the scores of big corporations around the country that pledged an array of money and initiatives toward racial justice amid the upheaval that Mr. Floyd’s death while in police custody unleashed.

Now, business leaders are facing new pressures from progressive activists to prove that those commitments were more than just talk. As activists press companies to condemn new voting legislation, CEOs are again finding themselves walking a difficult line on emotional, political issues, risking blowback from all sides.

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