Greater Leadership Overseas for the Benefit of Equality Act of 2019 or the GLOBE Act of 2019 This bill contains provisions related to (1) the protection of lesbian, gay, bisexual, transgender, and intersex (LGBTI) individuals globally; and (2) immigration protections for vulnerable individuals. The bill establishes (1) an interagency group to respond to urgent threats to LGBTI people in foreign countries, and (2) a permanent Special Envoy for the Human Rights of LGBTI Peoples in the Department of State. The President shall impose visa-blocking sanctions on foreign persons responsible for violating the human rights of individuals due to actual or perceived sexual orientation, gender identity, or sex characteristics. The State Department shall establish the Global Equality Fund to provide assistance to nongovernmental organizations working to advance and protect human rights. The U.S. Agency for International Development shall establish the LGBTI Global Development Partnership to work with the private sector and nongovernmental organizations to support LGBTI rights. The bill removes prohibitions against providing foreign assistance to nongovernmental organizations that offer certain medical services using non-U.S. government funds. The bill contains several immigration-related provisions, such as those (1) establishing that persecution based on sexual orientation or gender identity may form the basis of an asylum request, (2) removing the general requirement that an asylum claim must be filed within one year of the applicant's arrival in the United States, (3) establishing that a permanent partnership between two partners is a marriage for immigration law purposes, and (4) establishing that indigent aliens have a right to free counsel in certain immigration proceedings.