I can not find any prohibition against a US FAA certificated flight instructor giving instruction in a foreign country for a US FAA issued pilot certificate or endorsements. Title 14 of the Code of Federal Regulations Part 61.51 specifically allows a CFI to give instruction in a foreign aircraft registered in a country that is a member state of ICAO. Part 61.41 specifically allows a foreign CFI from a country that is a member state of ICAO to give instruction.
So, this implies that a US CFI can give instruction in a foreign country for a US FAA issued pilot certificate or endorsements. Just as long as the CFI has authorization to fly the foreign registered aircraft in the country of question. Depending on the country, they may need authorization to fly for hire. Depending on the country, they may need authorization to instruct as well. This may only apply to instructing for the issuance of a certificate or endorsement for that country.
So, at the very least, instructing in an aircraft of foreign registry in a foreign country for a US certificate may only require foreign validation of the CFI’s pilot certificate. But, maybe not their instructor certificate.
§61.41 Flight training received from flight instructors not certificated by the FAA. (a) A person may credit flight training
toward the requirements of a pilot certificate or rating issued under
this part, if that person received the training from:
(1) A flight instructor of an Armed Force in a program for training
military pilots of either—
(i) The United States; or
(ii) A foreign contracting State to the Convention on International
(2) A flight instructor who is authorized to give such training by the
licensing authority of a foreign contracting State to the Convention
on International Civil Aviation, and the flight training is given
outside the United States.
(b) A flight instructor described in paragraph (a) of this section is
only authorized to give endorsements to show training given.
§61.51 Pilot logbooks. (j) Aircraft requirements for logging flight time. For a person to log flight time, the time must be
acquired in an aircraft that is identified as an aircraft under
§61.5(b), and is—
(1) An aircraft of U.S. registry with either a standard or special
(2) An aircraft of foreign registry with an airworthiness certificate
that is approved by the aviation authority of a foreign country that
is a Member State to the Convention on International Civil Aviation
(3) A military aircraft under the direct operational control of the
U.S. Armed Forces; or
(4) A public aircraft under the direct operational control of a
Federal, State, county, or municipal law enforcement agency, if the
flight time was acquired by the pilot while engaged on an official law
enforcement flight for a Federal, State, County, or Municipal law
§61.195 Flight instructor limitations and qualifications. A person who holds a flight instructor certificate is subject to the
(a) Hours of training. In any 24-consecutive-hour period, a flight
instructor may not conduct more than 8 hours of flight training.
(b) Aircraft Ratings. Except as provided in paragraph (c) of this
section, a flight instructor may not conduct flight training in any
aircraft unless the flight instructor:
(1) Holds a flight instructor certificate with the applicable category
and class rating;
(2) Holds a pilot certificate with the applicable category and class