Yes - the Korea War is when the UFP is introduced as a security mechanism for the General Assembly (via R377). UFP is then activated in 1951 (R498) to lend further legitimacy to the United Command, while in effect influencing the war effort to a limited extent.

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UFP is powerful on the surface, yet in reality is still contingent on the backing / political will of superpowers USA USSR. Even when SC was deadlocked and UFP was activated, it could not influence political development on the ground if it infringes upon the geopolitical designs of the superpowers.

- Example, 1956 USSR Invasion of Hungary - UFP R 1004 was unable to force USSR to withdraw troops from Hungary.

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Also, the Security Council recognised that UFP could shift the primacy of the UN's security role away from the SC to the GA, hence over the years resorted less to stalemated situations that allowed the GA to fill the political void, all to preserve their primary role in resolving international disputes.

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Side-note - member states (e.g. USSR) also questioned the legality of UFP, since it expands GA's power on security issues, a mechanism that was clearly not dictated in the Charter - hence arguably violates UN's sovereignty principle when UFP resolutions order interventions in countries without their explicit permission, like in Korea.

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Definitely not the best answer but hope it helps a bit!

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FYI:

R377: Confers a subsidiary responsibility for maintaining international peace & security (IPS) on the GA

"If the SC fails to exercise its primary responsibility for the maintenance of int’l peace and security… the GA shall consider the matter immediately with a view to making appropriate recommendations to Members for collective measures, including in the case of a breach of the peace or act of aggression the use of armed force when necessary, to maintain or restore IPS.

If not in session at the time, the GA shall therefore meet in emergency special session within twenty-four hours of the request."

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R 498 (1951) urged states to continue furnish military assistance to Korea (after China entered the war)