What was the immediate significance of the Battle of Dunkirk on international relations?
American and Britain were already close diplomatically by the time of the Dunkirk miracle in May and June of 1940, but America was still isolationist and would not support our entry into World War II against Germany at that time and FDR, knowing this, also had to run for re-election in the fall.
So the immediate effect of the battle, which, despite the successful evacuation of British and allied troops from the port of Dunkirk, was still a horrible defeat, was to lend a sense of urgency to FDR's efforts to help Britain militarily with whatever support was possible. He was able to convince Congress to amend the 1936 Neutrality Acts and allow 50 World War I era destroyers to be transferred to the British in exchange for leases on nine military bases in the Western Hemisphere.
The British badly needed the ships to protect convoys of supplies from America and to protect the shores of England against seaborne invasion.