In 2012 Barclays paid $450 million to settle rate-rigging allegations. Bloomberg News

Across the globe, efforts to crack down on bribery have generally gone more slowly than they have in the U.S., where stepped-up enforcement of a 1977 law has in recent years led to more prosecutions and tens of billions of dollars in fines.

But the gap between the U.S. and the rest of the world may soon start to shrink.

The U.K., China, Brazil and Canada, seeing the money the U.S. has collected going after companies with antibribery investigations and prosecutions, have all enacted their own antibribery laws within the past few years.

Now the nations are starting to get their own investigations under way, say lawyers and antibribery experts.

"We haven't seen many enforcement examples yet...but they are certainly what's next, and that's what financial institutions should be worried about," said Kelly Kramer, a partner at Mayer Brown in Washington, D.C., last week during a session hosted by the firm on emerging issues in international anticorruption enforcement.

In creating their anticorruption statutes, several countries looked to the Foreign Corrupt Practices Act. The U.S. law, passed 37 years ago, makes it illegal to offer money or a gift to foreign-government officials or employees to gain a business advantage.

Of specific importance to companies considering cross-border mergers: provisions that allow for prosecutions against companies that make bribes anywhere in the world if they do business in the country that passed the law.

The rule, incorporated into the laws of the U.K. and Canada, requires companies considering tie-ups to conduct exhaustive due diligence, said Laurence Urgenson, a leader of Mayer Brown's anticorruption and Foreign Corrupt Practices Act practice.

After the Canadian Corruption of Foreign Public Officials Act was passed in 1999, there was confusion as to who would take the lead in investigating and prosecuting cases, said John Boscariol, a partner at the Toronto-based law firm McCarthy Tetrault LLP.

But amendments to the law passed last year gave the Royal Canadian Mounted Police exclusive authority to investigate and seek prosecutions.

So far, Canadian authorities have notched two convictions of companies under the CFPOA, and landed a three-year prison sentence in May for a man who played a central role in a bribery case, Mr. Boscariol said.

"Canada has finally picked up its game," he said, adding there are several dozen investigations currently under way. A spokesman for the RCMP couldn't be reached for comment.

The U.K.'s Bribery Act took effect in 2011, and major cases should start coming forward soon, said Robert Barrington, managing director of the U.K. office for Transparency International, an organization devoted to rooting out corruption around the world.

Mr. Barrington cited investigations by the U.K.'s Serious Fraud Office of Barclays PLC and GlaxoSmithKline PLC as examples of the agency stepping up its efforts.

GlaxoSmithKline is being probed for possible criminal violations in its commercial practices. The company said it was cooperating with investigators.

The SFO in April charged three former employees of Barclays for their alleged roles in manipulating the London interbank offered rate, or Libor. Lawyers for the men have either denied wrongdoing or declined to comment. In 2012 Barclays admitted wrongdoing and paid $450 million to settle rate-rigging allegations by U.S. and U.K. authorities.

While China has been aggressive in initiating anticorruption prosecutions, Mr. Barrington said he's wary of whether those actions are more about politics than cleaning up corruption, especially since China hasn't really cracked down on Chinese companies doing business internationally. "Can a country really be claiming to take corruption seriously if it's not controlling its own companies?" he asked.

So far, China has targeted pharmaceutical companies doing business in the country, including GlaxoSmithKline.

GlaxoSmithKline, which has said it is cooperating with Chinese authorities, has since overhauled its operations in China and announced a global policy to stop paying doctors to attend medical meetings or to speak about its drugs.

Write to Ben DiPietro at Ben.Dipietro@wsj.com