Agents stopped two of the vehicles and made multiple arrests, the statement said, but the Suburban drove away.
He said his deputies are often involved in chases pursuing people suspected of smuggling undocumented immigrants and drugs.
"From what we can tell the vehicle ran off the road and caught gravel and then tried to recorrect", Boyd said, adding that "caused the vehicle to turn over several times".
The chase had reached speeds topping 100 miles per hour, the sheriff added.
Critics of the immigration policies say that the American immigration officials have shown no plan that outlines the methods which will be used to reunite the families once the parents are persecuted for their "illegal border crossing". Another occupant remains in critical condition with potentially life-threatening injuries.
Boyd said the driver of the Suburban was also injured and was flown to a San Antonio hospital.
"It's a problem. It's a ideal example of why our borders need to be secure", Dimmit County Sheriff Marion Boyd told KSAT-TV. ProPublica said the audio features 6-year-old Alison Jimena Valencia Madrid who was separated from her mother last week at a border facility, as well as other children, border patrol agents and consular workers. Several people absconded from the vehicle.
While journalists were allowed to briefly tour the McAllen, Texas facility on Sunday, officials did not allow them to take pictures, shoot any video or interview any of the more than 1,100 people who were waiting inside. A fifth died later at the hospital.
We were told by the sheriff's office all of the immigrants inside the SUV were from Mexico, but the Mexican Consulate says they can only confirm that a majority are from Mexico.
One of the drivers attempted to flee the scene and was chased by agents, US Border Patrol said in a statement.
Since the policy was brought in almost 2,000 children are thought to have been taken away from their parents because they crossed the border and were living in the U.S. illegally.
"What's missing is a conductor", says the man in Spanish, whom ProPublica identifies as a border agent.
If you've been paying any attention to the news in recent days, you've no doubt heard something about the roiling controversy surrounding the Trump administration's terrible immigration policies and the frightening impact it's been having on the families it's been destroying.
"I think we need a wall, in my opinion", he said, voicing support for President Donald Trump's proposal to construct a border wall with Mexico.