COLUMBUS, Ohio — Companies that make and distribute opioids didn’t abide by a requirement that they refuse to ship orders of the powerful prescription painkillers when they deemed them suspicious, helping fuel a national addiction and overdose crisis, two Ohio counties said in a legal filing Friday.

Until now, lawyers representing Cuyahoga and Summit counties, the first local governments in line for a trial in a massive series of lawsuits seeking to hold the drug industry accountable for the crisis, have been focusing largely on allegations that drug companies made false claims about the safety of opioids, encouraging doctors to prescribe the drugs at higher doses and for more patients.

The latest filings, which came as part of a flurry of motions from both sides in the case, shifted the focus to whether companies complied with Drug Enforcement Administration requirements about how the drugs flowed to distributors and pharmacies as the death toll from overdoses of prescription and illicit opioids rose.

One executive at Mallinckrodt emailed a distributor requesting he check the inventory of a drug. “If you are low, order more,” Victor Borelli wrote, according to the document. “If you are okay, order a little more.”

The filing said that, from 2003 through 2011, the maker of generic drugs shipped 53 million orders of opioid painkillers and flagged 37,817 as “peculiar” — but withheld just 33 of those.

The plaintiffs say DEA policy required that suspicious orders be reported and not shipped unless it was determined they were not likely to be diverted. The interpretation of that policy is in dispute in the case.

In an email, Mallinckrodt spokesman Jeffrey Taufield called the communication “an outrageously callous email from an individual who has not been employed by the company for many years.”

“It is antithetical to everything that Mallinckrodt stands for and has done to combat opioid abuse and misuse,” he added.

The plaintiffs zeroed in on other companies, including Teva, which they said had never had a written system for monitoring suspicious orders as of at least September 2012 — and had never reported a suspicious order to the DEA by then.

An email seeking comment was sent to a spokeswoman for the company Friday night. An email from the company earlier in the week said that “Teva has not conspired, failed to report suspicious orders or contributed to the abuse of opioids in the U.S. We maintain a comprehensive and robust system to prevent suspicious orders from ever entering the market.”

The allegations come days after a key set of data maintained by the federal government was made public in the cases. According to a Washington Post analysis, it shows that 76 billion pain pills were shipped from 2006 through 2012, a span when overdose deaths ballooned. According to federal data, opioids — both prescription drugs such as Vicodin and OxyContin and illicit versions of fentanyl and illegal heroin — were factors in more than 400,000 deaths in the U.S. from 2000 through last year.

More than 2,000 state, local and tribal governments have filed similar claims against the drug industry. Most of them have been consolidated under Cleveland-based U.S. District Judge Dan Polster, who has scheduled the trial for the two Ohio counties’ claims to start Oct. 21. The state-court opioid trial of a suit brought by Oklahoma against Johnson & Johnson wrapped up earlier this week, but the judge in it has not yet given a verdict.

Parties in the Ohio case asked the judge to rule on various claims without having to go to trial.

Most of the filings Friday came from defendants seeking to have pieces of the plaintiffs’ claims rejected. Drug distributors say there’s no evidence they conspired to weaken regulations or increase federal limits on how many opioids could be made.

Drugmakers and distributors said there’s no evidence that they illegally pushed unnecessary prescriptions that led to a drug crisis.

Makers of generic opioids, for instance, argued they can’t be sued for making negligent marketing claims because they do not make claims about the safety or efficacy of drugs. And drugmakers moved to exclude testimony that asserts prescription painkillers were a gateway to street drugs such as heroin and fentanyl, calling the idea “too speculative and unreliable as a matter of science, and too remote as a matter of law.”

Defendants asked that jurors not hear any expert testimony on how much it would cost to abate the opioid crisis. According to the filings, one Johns Hopkins University study pegged the cost at as much as $453 billion over 10 years with the biggest portion going to addiction treatment.

The companies also said that claims about anything before October 2012 should be dismissed because of statute of limitation laws.

Still other filings were from companies such as Walmart arguing to be dropped as defendants entirely and asking the court to exclude testimony from people that parties want to use as expert witnesses and from pharmacies and small distribution companies that argue that conspiracy claims against them cannot be proven.

Mulvihill reported from Cherry Hill, N.J.