A seven-year-old Texas boy has died after his father's girlfriend allegedly opened fire as the man drove away with his son and two other children, police said.

Greenville police said Kaden Green died at a hospital after being taken there by a witness following the shooting on Thursday afternoon.

The woman accused of shooting the boy, 26-year-old Brooke Craig, was arrested early on Friday along with the child's father, 24-year-old Cameron Castillo, in the Dallas suburb of Frisco about 45 miles west of Greenville.

Authorities said Craig led police on a car chase before their arrests.

Craig faces a capital murder charge, while Castillo was arrested for a parole violation.

Officers were responding to a report of a disturbance at a home on Thursday when they received a call that a boy had been taken to a hospital with a gunshot wound to the chest.

Police said they believe Castillo and Craig had been involved in a disturbance at the home before he got into his vehicle and headed down the street.

After the shooting, the vehicle came to a stop and Craig and a witness approached it, police said. The witness then took the boy to the hospital in another vehicle.

Police said the two other children who were in the vehicle were placed in the custody of the state.

Child Protective Services (CPS) spokeswoman Marissa Gonzales said they have a four-year-old boy and seven-year-old girl who are the siblings of Kaden in their custody.

Ms Gonzales said Craig is not Kaden's mother. She said CPS has received a report about the boy's death and will investigate. She said the boy's family has had previous contact with CPS, but that the details are confidential.

McKinney police said Greenville police contacted them early on Friday about the capital murder warrant for Craig.

McKinney police then located her vehicle leaving a store and she fled into nearby Frisco, going through several residential areas before turning down a dead-end street, police said.

Craig and Castillo, who was her passenger, were arrested without incident after their vehicle struck a police cruiser while attempting to back up, police said.

Jail records show that Craig was transferred from Collin County, where McKinney is located, back to Hunt County, where Greenville is located, by Friday afternoon. The records did not list a lawyer in relation to the capital murder charge.

Craig faces an evading arrest charge in Collin County, according to the jail records.

Castillo was being held in Collin County jail on Friday. Jail records did not list a lawyer for him.