When Michael Sabbie was booked into jail by the Arkansas police on a misdemeanor assault charge in July 2015, he warned nurses there about his ailments — heart disease, high blood pressure and asthma — and told them he needed medication.

Less than three days later, Mr. Sabbie was dead on the floor of his cell. Videos captured his rapidly deteriorating health in the hours before his death as he pleaded with corrections officers for help. At least 19 times he could be heard saying, “I can’t breathe” — at one point as he crawled, gasping for air, while guards watched him through his cell door.

A federal lawsuit filed by his family on Wednesday accuses at least 12 corrections officers and nurses at a for-profit jail on the Texas-Arkansas border of causing his death. The lawsuit claims that the employees at the jail, the Bi-State Justice Center in Bowie County, Tex., showed a “deliberate indifference” to his health and ignored obvious signs of his declining condition.