One of the four taken to the San Antonio hospital later died, the department said.
Sheriff Boyd said there needs to be a wall built around the area because it's a "real problem". That agent stopped one vehicle and radioed a description of the other two vehicles to other agents, the agency said.
Boyd said a total of 14 people were inside the SUV and several were ejected out of the vehicle when it rolled onto Highway 85.
Most of the occupants in the SUV were believed in the country without legal permission.
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Border Patrol officials have repeatedly warned of the dangers of smuggling large numbers of people in vehicles.
The incident comes amid heightened tensions over the treatment of immigrants at the southern border. Officials said they believe the driver and passenger are both USA citizens. "This is I think a ideal example of why our borders need to be secured".
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He also spoke of how happy he was that Trump had agreed to meeting with Kim, which no USA president has done before. The meeting was the first between a sitting US president and a North Korean leader.
The Trump administration has said tougher immigration policies are needed to deter immigrants from entering America illegally.
Some injured were taken by helicopter to San Antonio, about 90 miles northeast.
In a later interview, Boyd said the auto reached 100 miles per hour before the driver lost control and crashed. Dimmit County is directly north of Webb County and east of Maverick County, which border Mexico.
The sheriff said he hadn't spoken to Border Patrol on the exact cause of the pursuit and nature of the crash.
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Giuliani, who is an attorney for the president, recommended the Trump administration to adopt the same approach with Palestinians. The president said he was also optimistic about a normalisation of diplomatic relations with North Korea in the long term.