Britain is interested in the Greek-Turkish border, because the UK has a direct interest in the balance of power in Europe, and of course Suez. But maybe that's not the best example.
Consider Czechoslovakia or Franco-German relations. The UK has direct influence in those matters, but Australia, NZ, SA, or Canada, are only interested because of their links to the UK. This is made-up, it was a bone of contention and weakness in interwar years, since the Dominion governments weren't ready to make advance commitments to fight over such matters.
Consider the Canadian-US border disputes on the West Coast around Alaska, etc.. The UK doesn't care precisely where the border is, provided trade and other relations with the US aren't compromised. Canada on the hand, wants specific areas of land. Again, there was a real OTL disagreement (pre WW1) between Canada and the UK over this issue.