What is the process of deposing an Orthodox Patriarch as head of their particular Orthodox Church?

Autocephaly is the status of a hierarchical Christian church whose head bishop does not report to any higher-ranking bishop. The term is primarily used in Eastern Orthodox and Oriental Orthodox churches.

One question that needs to be answered is whether the Ecumenical Patriarchate of the Eastern Orthodox Church has authority over the Moscow Patriarchate.

The following Wikipedia article suggests that the power to grant autocephaly is in dispute. https://en.wikipedia.org/wiki/Autocephaly#Autocephaly_in_Eastern_Orthodoxy

In Eastern Orthodoxy, the right to grant autocephaly is nowadays a contested issue, the main opponents in the dispute being the Ecumenical Patriarchate, which claims this right as its prerogative, and the Russian Orthodox Church (the Moscow Patriarchate), which insists that one autocephalous jurisdiction has the right to grant independence to one of its components. Thus, the Orthodox Church in America was granted autocephaly in 1970 by the Moscow Patriarchate, but this new status was not recognized by most patriarchates. In the modern era, the issue of autocephaly has been closely linked to the issue of self-determination and political independence of a nation; self-proclamation of autocephaly was normally followed by a long period of non-recognition and schism with the mother church.

This seems to be the situation with regard to the Patriarchate of Moscow and the Kyiv Patriarchate:

https://orthodoxwiki.org/List_of_autocephalous_and_autonomous_churches There are two ways to make a list of autocephalous and autonomous churches. The first way is to list only the jurisdictions that are universally recognized as self-governing. This may be called the "minimal list", and it is used in most inter-Orthodox gatherings. The second way is to list all those jurisdictions that are recognized as self-governing by at least one other jurisdiction.

The Patriarchate of Moscow is recognized as autocephalous or autonomous by all the others. The Orthodox Church of Ukraine is autocephaly recognized by Constantinople, Alexandria, Cyprus and Greece. The Church of Ukraine (Kyiv Patriarchate) is in schism from the mainstream Orthodox Church. In other words, they are not in communion and not considered part of the Orthodox Church, although in most cases they maintain the same doctrine and the same practices.

This is how the Moscow Patriarchate views the independent Kyiv Patriarchate:

https://en.wikipedia.org/wiki/Autocephaly#Modern-era_historical_precedents The Act of Declaration of Independence of Ukraine was adopted by the Supreme Soviet of the Ukrainian SSR on 24 August 1991. The Act reestablished Ukraine's state independence. The independent Kyiv Patriarchate was proclaimed in 1992, shortly after the proclamation of independence of Ukraine and the dissolution of the USSR in 1991. The Moscow Patriarchate has condemned it as schismatic, as it claims jurisdiction over Ukraine. Some Orthodox churches have not yet recognized Ukraine as autocephalous. In 2018, the problem of autocephaly in Ukraine became a fiercely contested issue and a part of the overall geopolitical confrontation between Russia and Ukraine, as well as between the Moscow Patriarchate and the Ecumenical Patriarchate of Constantinople.

In Eastern Orthodoxy, autonomy denotes a type of limited self-government of a church which still depends to some degrees on its mother church. A church that is autonomous has its highest-ranking bishop, such as an archbishop or metropolitan, approved (or ordained) by the primate of the mother church, but is self-governing on some aspects; the aspects on which the autonomous church is self-governing depends on the decision of the mother church.

Is there any authority higher than the Moscow Patriarch? Or is he so powerful that he cannot be deposed? Would any group of men dare to stand up to him? I regret to say I simply don't know, but the heirarchical structure suggests that the Moscow Patriarch is fireproof.