Missing Persons and Unidentified Remains Act of 2019 This bill revises the authority for state grants to improve the reporting of unidentified and missing persons. The bill authorizes grants to be used for additional activities. In particular, grants may be used to improve the transportation, processing, and identification of missing persons and unidentified remains, including with respect to migrants. Further, the bill expands eligible entities to include local governments, medical examiners' offices, certain accredited and publicly funded laboratories, and certain nonprofit organizations. In addition, the bill expands the allowable uses of grant funds to include costs related to transportation, hiring, and procuring forensic equipment. The bill requires any DNA reference sample that is received from a suspected biological family member and uploaded into the Combined DNA Index System to be used only for identifying missing persons and unidentified remains. Next, the bill requires the U.S. Customs and Border Protection (CBP) to purchase, deploy, and maintain no more than 170 rescue beacons along the southern border to mitigate migrant deaths. The bill also requires the CBP, the Department of Justice, and the Government Accountability Office to submit specified reports to Congress.