A Connecticut judge, exasperated by the Infowars broadcaster Alex Jones’s “bad faith” failure to sit for a deposition in a lawsuit brought by the families of Sandy Hook Elementary School shooting victims, ruled on Wednesday that he would be subject to escalating daily fines for future delays.

The judge, Barbara Bellis of Connecticut Superior Court, found Mr. Jones in contempt and ordered that he be fined $25,000 for the first weekday he fails to appear for testimony, beginning on Friday. For every day thereafter that he does not appear, the daily fine will increase by $25,000. She also ordered that he be deposed in Connecticut, rather than in his home of Austin, Texas.

If Mr. Jones fails to testify by April 15, Judge Bellis will impose further sanctions, potentially revoking his ability to call witnesses or present evidence in the trial. The judge rejected a motion by the families’ lawyers that Mr. Jones be jailed until he testifies.

Mr. Jones for years spread bogus claims that the Dec. 14, 2012, shooting of 20 first graders and six educators in Newtown, Conn., was staged by the federal government as a pretext for confiscating Americans’ firearms, and that the families were actors in the supposed plot. The falsehoods have led to years of torment and threats against the victims’ relatives.

The families of 10 Sandy Hook victims are suing Mr. Jones for defamation in three separate lawsuits: two in Texas filed by the families of two victims and one in Connecticut filed by the families of eight victims and an F.B.I. agent targeted by conspiracy theorists.

Over nearly four years of litigation with the Sandy Hook families, Mr. Jones repeatedly delayed proceedings, violated court rules and failed to submit business records and testimony ordered by the courts. Late last year, judges in Texas and Connecticut ruled him liable by default, granting the families a sweeping victory. In trials slated to begin next month in Texas, juries will determine how much Mr. Jones must pay the families in damages. The Connecticut trial is the last scheduled trial, set to begin on Sept. 1.

With the trials looming, Mr. Jones continues to stonewall. Judge Bellis last week rejected his claim that a medical emergency prevented him from sitting for a two-day deposition in Austin. The day before his court-ordered deposition, Mr. Jones was broadcasting from his Infowars studio while his lawyer, unknowing, told the judge he was under a doctor’s care at home.

“The plaintiffs subjected themselves to hours and hours of painful questioning by Mr. Jones’s lawyers — and Mr. Jones plays sick when it is his turn to tell the truth under oath,” the families’ motion for contempt read.

On his show that day, Mr. Jones falsely cast the proceedings as part of a plot by Democrats to ensnare him, and pleaded for listeners to send money for his legal expenses. Records released in the Texas cases indicate Infowars had revenue of more than $50 million annually during the years Mr. Jones — who sells diet supplements, survival gear and other merchandise while broadcasting conspiracy-themed content on the radio and online — was a vocal supporter of President Donald J. Trump.

The House committee investigating the Jan. 6 attack on the Capitol has subpoenaed Mr. Jones, who echoed Mr. Trump’s lie that the 2020 election was stolen, and is scrutinizing his role in organizing “Stop the Steal” events.

Late Tuesday, Mr. Jones’s lawyers filed an “offer of compromise” in Connecticut, proposing to pay the Sandy Hook plaintiffs a settlement of $120,000 each. “Mr. Jones extends his heartfelt apology for any distress his remarks caused,” they said.

The families swiftly rejected the proposal. “The so-called offer is a transparent and desperate attempt by Alex Jones to escape a public reckoning under oath with his deceitful, profit-driven campaign against the plaintiffs and the memory of their loved ones lost at Sandy Hook,” they said in a statement.

The families in the Texas cases, in which Mr. Jones has been deposed, also say they are eager for their day in court.

“I’m looking forward to staring him down in a courtroom, to remind him of how he threw salt in my wound,” Lenny Pozner, whose son Noah died at Sandy Hook, said in an interview. Mr. Pozner and his nonprofit, the HONR Network, have led efforts to hold Mr. Jones to account for his falsehoods about the shooting. “He’s been hung by his own actions,” Mr. Pozner added.

Chris Mattei, a lawyer on the families’ legal team in Connecticut, would depose Mr. Jones. Mr. Mattei said the lawyers also plan to depose Rob Dew, a top Jones lieutenant, and finish questioning several Infowars representatives. Mr. Jones’s father, David Jones, who bankrolled his son’s business at its founding and is deeply involved in its product sales, has so far evaded a subpoena seeking his deposition, Mr. Mattei said.

Of Mr. Jones’s many maneuvers in the run-up to trial, Mr. Mattei said the most egregious was “his fabrication and concealment of financial records” and failure to produce comprehensive data on traffic to the Infowars website.

“Our view has long been that one of the reasons Jones lies the way he does is because he is able to provoke in his audience the fear, resentment and distrust that make them easy marks to sell stuff,” Mr. Mattei said in an interview.

“We have claimed that the financial records, in combination with web traffic data, would show exactly that — that he has profited from false claims that Sandy Hook is a hoax,” Mr. Mattei said.