Dallas County Schools fired a school crossing guard only eight days after she was hired when a 40-year-old arrest emerged on her background check.

In the decades since Beverly Harrison’s 1975 arrest for aggravated assault, the then-57-year-old had never again been convicted of a crime. Before she was fired in 2013, she had previously spent 30 years working for the city of Dallas, and later as a home health aide and a cafeteria worker.

Advocates say the black grandmother’s story is indicative of the consequences for millions of people in Texas, particularly people of color, if Texas prevails in its...