Mr. Jones peddles diet supplements, survivalist gear and gun-related paraphernalia on radio broadcasts and videos that spread outlandish claims like the government is trying to infringe on Americans’ rights, destroy their health or control their minds.

On Tuesday, Twitter suspended Mr. Jones for a week after he posted a link to a video calling for supporters to get their “battle rifles” ready for a fight against the press and others, violating the company’s rules against inciting violence.

Also this week, the Federal Communications Commission shut down a pirate radio station that served as Infowars’ flagship outlet, and which has operated without a federal license since at least 2013, The Austin American-Statesman reported.

Friday’s motion is the latest legal salvo in three separate defamation lawsuits filed by Sandy Hook families, which seek tens of millions of dollars in damages and pose an existential threat to Mr. Jones’s business. Should the court find that Mr. Jones and Infowars willfully destroyed evidence, he, and possibly his lawyer, could be assessed thousands of dollars in fines and be subject to punitive action. Most important, the material that was destroyed could be presumed by the court as supporting Mr. Heslin’s claims against Mr. Jones, bolstering his case.

Besides the two cases in Texas, the families of seven more Sandy Hook victims and an emergency medical worker subjected to harassment filed a separate defamation suit against Mr. Jones and his associates in May in Connecticut. The families in the larger case are represented by Koskoff, Koskoff & Bieder, a Bridgeport, Conn., firm that also represents Sandy Hook families in a lawsuit against Remington, the maker of the AR-15-style weapon used in the shooting.

The first court appearance in the Sandy Hook lawsuits was in Texas this month, when the court heard arguments in Mr. Jones’s motion to dismiss the defamation case brought by Leonard Pozner and Veronique De La Rosa, the parents of Noah Pozner, a 6-year-old killed at Sandy Hook. A decision is expected early next month. A hearing is scheduled for Aug. 30 in Mr. Jones’s motion to dismiss the second Texas case, brought by Mr. Heslin.

A ruling on Friday’s motion alleging destruction of evidence is expected before the Aug. 30 hearing.