The US Foreign Affairs Manual addresses this question at 7 FAM 086.
It acknowledges the well established rule that a dual citizen of A and B cannot rely on obtaining consular assistance from A when held in custody in B (though B might choose allow consular access as a courtesy). The language about a dual citizen incarcerated in a third country is significantly different, and suggests that there is no general rule or principle of international law governing such cases.
On the first subject, namely that of a US dual national arrested in the other country of nationality (which is not the subject of this question, but is quoted for contrast and comparison):
b. It is a generally recognized rule, often regarded as a rule of international law, that when a person who is a dual national is residing in either of the countries of nationality, the person owes paramount allegiance to that country, and that country has the right to assert its claim without interference from the other country. Thus, in the absence of agreements to the contrary between the United States and the country of second nationality, if a dual national encounters difficulties in the country of the second nationality while residing there, the U.S. government's representations on that person's behalf may or may not be accepted.
c. Dual nationals traveling abroad on a passport of their other country of nationality (or even on a U.S. passport in the absence of an express agreement between the United States and the other country) may find that the host country treats them as a national of only that country and does not recognize the United States as a country entitled to provide consular services. Nonetheless, you should provide consular services to the fullest extent permitted by the host country.
It goes on to say
d. When a U.S. citizen is a dual national, but is not a citizen of the host country, and the second country of nationality is providing protective services to the dual national, you should consult with the citizen and your foreign consular colleagues to ensure that appropriate protection is provided. The host country might permit consular visits only by the representative of one country, often the nation on whose passport the citizen entered the foreign country. ...
e. The United States does have a limited number of bilateral consular agreements or arrangements that address certain questions of related to dual nationals and consular assistance. These agreements generally provide that all nationals of the United States entering the other country on the basis of U.S. travel documents containing properly executed entry and exit visas will, during the period for which their status has been accorded, and in accordance with the visa's period of validity, be considered nationals of the United States by the appropriate authorities of the foreign country for the purpose of ensuring consular access and protection by the United States. ...
There is also a link to 7 FAM 416.3:
7 FAM 416.3 Dual Nationality
Providing consular protection to dual nationals sometimes poses complex problems because of the conflicting laws and regulations of the United States and other countries. Consular officers are required to open a case, file an arrest report and update the Department on your efforts to secure access.
7 FAM 416.3-1 Dual National Arrestees In The Non-U.S. Country Of Nationality
a. The most complex problems regarding provision of consular services to dual nationals arise when the holder of dual nationality is arrested in their other (non-U.S.) country of nationality. While consular officers do not usually have a right to consular access to a dual national detained in one of their countries of nationality, attempts should still be made to seek consular access on a courtesy basis from the government of the receiving state.
b. See the Consular Notification and Access Manual for information on dual nationals detained in the United States and Department of State instructions to law enforcement with respect to them.
There is nothing in this section on dual nationality that discusses the case of a US dual national detained in a third country.
You write:
This is about whether country C is obligated to allow B's assistance, or is it only bound by the nationality declared at the point of admission by presenting passport A?
The answer seems to be that this is not settled in international law and that different countries may take different approaches to the question. 7 FAM 086 does have some examples of bilateral agreements, but these concern people arrested in the other country of nationality, not in a third country. The bilateral agreements mentioned in fact override the default rule and allow a dual national to obtain consular assistance if they entered one country of nationality using documents issued by the other:
U.S. – China Bilateral Consular Convention – The Exchange of Notes appearing at the end of the text of the Convention provide:
“2. The two governments agree to facilitate travel between their respective countries of persons who may have a claim simultaneously to the nationality of the United States of America and the People's Republic of China, but this does not imply that the governments of the two countries recognize dual nationality. Exit formalities and documentation shall be dealt with in accordance with the laws of the country in which such person resides. Entry formalities and documentation shall be dealt with in accordance with the laws of the country of destination.
“3. All nationals of the sending State entering the receiving State on the basis of travel documents of the sending State containing properly executed entry and exit visas of the receiving State will, during the period for which their status has been accorded, and in accordance with the visa's period of validity, be considered nationals of the sending State by the appropriate authorities of the receiving State for the purpose of ensuring consular access and protection by the sending State as provided for in Article 35 of the Consular Convention between the United States of America and the People's Republic of China. If judicial or administrative proceedings prevent the above-mentioned persons from leaving the country within the visa's period of validity, they shall not lose the right of consular access and protection by the sending State. Such persons shall be permitted to leave the receiving State without the necessity of obtaining documentation from the receiving State other than the exit documentation normally required of departing aliens.”
Some arrangements on this subject are less formal. For example:
(1) A 1994 agreement between the United States and Vietnam provides for immediate notification of and reciprocal access within 96 hours to each other's detained citizens. Bearers of U.S. passports who enter Vietnam with a Vietnamese visa, including those of Vietnamese origin, are regarded as U.S. citizens by the U.S. Government for purposes of notification and access. Therefore, U.S. citizens are encouraged to carry photocopies of passport data and photo pages with them at all times so that, if questioned by Vietnamese officials, proof of U.S. citizenship is readily available.
(2) The U.S.- DPRK Interim Consular Agreement provides that North Korea will notify the Swedish Embassy within four days of an arrest or detention of an American citizen bearing a U.S. passport who enter the DPRK with a DPRK visa and will allow consular visits within two days after a request is made by the Swedish Embassy.
