Livestock Disaster Relief Act This bill modifies the Livestock Forage Disaster Program (LFP) and the Emergency Livestock Assistance Program (ELAP), with a particular focus on addressing the needs of agricultural producers during droughts. (The Department of Agriculture carries out these programs to assist agricultural producers affected by natural disasters.) The changes include increasing the number of months a producer may receive LFP assistance and allowing transportation costs for feed and water as covered losses under ELAP.