British Mission Records indicate that Elizabeth had secured passage with her husband (John Given), children, and some of her husband's family on the ship Ellen, which was scheduled to set sail on 8 January 1851. However, the 1851 census indicates that she, her sister-in-law, and youngest son Joseph, stayed behind. She and Joseph didn't travel until three years later. They arrived in New Orleans on 2 May 1854 via the ship The John M. Wood, where her husband and son John Jr. were waiting for them. Joseph died sometime shortly after their arrival in New Orleans.
Elizabeth continued on the overland journey with her son, John Jr. Her husband is not seen on the records. It is likely he hired out as a teamster on this or another company.