Eight children were killed and two other people were gravely wounded in a shooting spree that spanned at least three locations in Shreveport, La., and ended with the gunman shot dead after a police chase on Sunday morning, the authorities said.

The gunman, Shamar Elkins, 31, had mental health problems and had recently expressed suicidal thoughts, family members said in interviews.

The children ranged in age from 3 to 11, officials said, and seven of the eight were Mr. Elkins’s. He shot them execution style, according to a spokesman for the Shreveport Police Department.