This is a vague word that generally means a request or statement of position from one government to another. When a government makes a request or a demand of another government, it is in the form of a démarche. (How each country specifically defines that word varies.)
From Protocol for the Modern Diplomat (PDF), a historical document published by the U.S. State Department:
a request or intercession with a foreign official, e.g., a request for support of a policy, or a protest about the host government’s policy or actions
The actual form is often written, often as a diplomatic cable. Sometimes they are delivered verbally, usually followed by some written materials.
Démarches are usually delivered to the ambassador of the target government.
It’s a French word which means “step.”
Why I Looked It Up
I remember a plot point in the first episode of the Jack Ryan series on Amazon where Ryan persuaded someone to deliver a démarche to Yemen to freeze a bank account.
(I wasn’t alone: Google Trends shows a big spike for searches for “demarche” at the exact time the first episode of the series aired.)
Later, in a book about the oil industry, I read of a night in 1973 when Leonid Brezhnev, who was the Soviet General Secretary at the time, was staying at Richard Nixon’s California estate. Brezhnev had the Secret Service wake Nixon up for an unscheduled three-hour meeting where Brezhnev argued for new diplomatic action in the Middle East.
Nixon and National Security Assistant Henry Kissinger thought Brezhnev’s strange démarche was a heavy-handed ploy to force a Midwest settlement on Soviet terms […]
That seems to be an odd usage of the word. Waking up the President in the middle of the night for an impromptu conversation is only “formal” in the sense that the two parties were high-level representatives of their governments. In other usages, démarches are depicted as formal, weighty, measured requests.