Families of the Sandy Hook school shooting victims have won a series of victories in their defamation suits against the conspiracy theorist Alex Jones that would open Mr. Jones’s business records to them and compel him to speak under oath.

Ten families are pursuing lawsuits against Mr. Jones over his role in spreading bogus claims about the shooting, including that the victims’ families were actors in a plot to confiscate firearms from Americans. The families have endured death threats, stalking and online abuse.

Mr. Jones, a far-right provocateur and the owner of Infowars, a radio show and website on which he sells diet supplements, survivalist gear and gun paraphernalia, has come under growing scrutiny over the past year and has lost access to much of his online audience. Facebook, Twitter, Apple and YouTube have all banned him, and a recent deal for his show to stream on Roku was revoked last month after public outrage.

The suits by the Sandy Hook families have advanced on several fronts in recent weeks.

A Texas judge on Jan. 25 ordered Mr. Jones and representatives of his company to submit to questioning by lawyers for Scarlett Lewis, the mother of Jesse Lewis, one of the 20 children killed at Sandy Hook Elementary School in Newtown, Conn., in 2012. The Texas judge also granted access to Mr. Jones’s relevant business records, and denied his lawyer’s motion to keep the records sealed.