Broussard married a veterinarian visiting Denmark, a Shiite Muslim from Iraq. She moved to Baghdad with him when she was 18 and lived there through 5 military takeovers and 15 other coup attempts during the 1960s. They immigrated to the United States, living in Iowa, Illinois and Missouri before settling in Baton Rouge, Louisiana where her husband was an assistant professor. Soon after the move to Louisiana, the couple divorced, and Broussard finished her education at Louisiana State University, earning a bachelor's degree in psychology and a master of fine arts. She then worked as an art therapist and teacher at several schools in the area.
Broussard joined the Unitarian Church in Baton Rouge, and eventually went into the ministry. She met her current husband, Bruce Broussard, and they moved to Chicago so Lone could earn her Masters of Divinity at Meadville Lombard Theological School. She served at several congretations before getting the call to be pastor at Unitarian Universalist Fellowship in Wilmington, North Carolina, where she served from 1992 until 2001. In 2001 she was called to the mid-size Valley Unitarian Universalist Church of Phoenix, Arizona, where she served until being called to Birmingham. She was installed February 27, 2011. Her husband Bruce died in 2012.
Broussard and her ministerial intern Gail Stratton performed fifteen same-sex weddings in Linn Park outside the Jefferson County Courthouse on February 9, 2015, following a federal court order striking down the state's restrictions against such unions.