Money Remittances Improvement Act of 2013 - Authorizes the Secretary of the Treasury, with respect to compliance with reporting requirements for monetary instrument transactions of certain financial institutions, to rely upon examinations conducted by the relevant state supervisory agency if: (1) the category of financial institution is required by state law to comply with federal requirements, or (2) the state supervisory agency is authorized to ensure that the financial institution complies with such federal requirements.

Empowers the Director of the Financial Crimes Enforcement Network (FinCen) to furnish research, analytical, and informational services to foreign regulatory agencies in the interest of detection, prevention, and prosecution of terrorism, organized crime, money laundering, and other financial crimes.