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The stocks of Facebook and Alphabet were both down slightly Monday following a report that the U.K. government plans to create a regulatory body to oversee and enforce content rules on their vast digital platforms.

The proposal, revealed in a policy paper published early Monday, would force big tech companies to take “reasonable and proportionate” action on potentially harmful content such as terrorist propaganda, disinformation, and cyberbullying.

The back story. The U.K. plan, which could give the regulatory body authority to levy civil fines, is part of a larger push by governments from Australia to Germany to hold companies responsible for written and video content that flows through their systems to billions of people. Calls for regulation have intensified since claims Russia abused social-media tools to influence the 2016 U.S. presidential election, and last month’s terrorist attack in New Zealand, which the attacker broadcast live over Facebook.

What’s new. Big tech companies have been taking steps to tamp down harmful content through use of technology such as artificial intelligence, and hiring binges of security personnel—but not enough for the U.K.

“The era of self-regulation for online companies is over,” Jeremy Wright, the U.K.’s digital secretary, said in remarks accompanying the proposal rollout.

Big tech appears to realize that. They have come out in support of regulation, with the caveat that any rules must be specific and not undercut free speech and business operations.

“New rules for the internet should protect society from harm while also supporting innovation, the digital economy and freedom of speech,” Rebecca Stimson, Facebook’s head of U.K. public policy, said in a statement. “These are complex issues to get right and we look forward to working with the Government and Parliament to ensure new regulations are effective.”

She pointed to Facebook Chief Executive Mark Zuckerberg’s op-ed piece in the Washington Post last month, in which he acknowledged the need for new internet regulation.

Claire Lilley, a public policy manager at Google, said the company looks “forward to looking at the detail of these suggestions and working in partnership to ensure a free, open and safer internet.”

Facebook stock (ticker: FB) has declined 0.6% to $174.73, while Alphabet stock (GOOGL) has dipped 0.1% to $1,209.98, and Amazon.com is off 0.4% at $1,828.41.

Looking ahead. Tech is saying all the rights things, perhaps, because it doesn’t have much of a choice.

Stateside, Facebook and Google face more immediate pressure: Officials from both companies are scheduled to appear before the House Judiciary Committee on Tuesday to field questions about the spread of white nationalism on their platforms.

In addition to Tuesday’s scheduled hearing, it faces mounting pressure—even threats—from presidential candidates like Sen. Elizabeth Warren (D., Mass.). She wants to break up the operations of Facebook, Google, and (AMZN), citing their immense influence and the need to “promote more competition.”

Write to Jon Swartz at jonathan.swartz@dowjones.com