Specifically, their diplomatic immunity and its interaction with segregation laws. I understand they don't want to make waves, but it seems like it would be impossible in the face of such vitriolic racism.
There were fascinating conflicts and confrontations on all sides. Let's start with the United States.
Until 1961, the U.S. State Department relied on its Office of Security to provide escorts for visiting diplomats and dignitaries. Before the 1960s, foreign visits weren't all that common. Transportation technology was one hurdle, and another was the simple fact that were many fewer nations in the world. The process of decolonization created a wealth of new nations in Asia and Africa, and these nations could — thanks to jet transportation — send their dignitaries around the world.
According to the official history of the Bureau of Diplomatic Security, President Kennedy "entertained more dignitaries and heads of state during his first two years in office than Roosevelt had in his 12 years in office or Truman had in eight years."
The 1961 Vienna Convention specifically ordered host countries to provide diplomatic security, but the U.S. government was slow to respond. As a result of this — coupled with the racist attitudes prevalent in and around Washington, D.C., — there were several incidents.
A Ghanian diplomat who traveled to Georgia to observe an election was "roughed up" by white supremacists. An Ethiopian diplomat received threatening phone calls and found the tires of his car flattened again and again. When he complained to D.C. police, they ignored his request for an investigation.
African diplomats had a hard time finding housing in Washington, thanks to the ability of homeowners and landlords to discriminate. The same was true for their staffs and families. Washington's Metropolitan Club granted free membership to ambassadors — but not to African and Asian ones.
Kennedy realized this was a huge problem, so Undersecretary of State Chester Bowles proposed the creation of the Special Protocol Service Section of the State Department's Office of Protocol. As the section's first chief, Pedro Sanjuan, said: "What affects one or more members of these groups is likely to have a strong influence on the opinions and attitudes of their governments."
Nevertheless, the creation of that section couldn't solve all the problems. When a restaurant denied service to the new ambassador from Chad, he returned to Chad and quit. The trip he was on — from New York City to Washington, D.C., had been to present his credentials to Kennedy.
Secretary of State Dean Rusk, in As I Saw It (p. 382), tells a story about an African delegate to the UN who was traveling to New York when his plane stopped in Miami.
"When the passengers disembarked for lunch, the white passengers were taken to the airport restaurant; the black delegate received a folding canvas stool in a corner of the hangar and a sandwich wrapped with waxed paper. He then flew on to New York, where our delegation asked for his vote on human rights issues. That same ambassador later became his country's prime minister. We learned later that his chronic bitterness toward the United States stemmed from that incident."
For more detail about what I've written above, I suggest Renee Romano's "No Diplomatic Immunity: African Diplomats, the State Department, and Civil Rights, 1961-1964" in the September 2000 Journal of American History; Mary Dudziak's Cold War Civil Rights: Race and the Image of American Democracy; and Timothy Maga's "Battling the 'Ugly American' At Home" in Diplomacy and Statecraft.
Now, let's look at the other side of the coin — specifically, when Edward Perkins was appointed the U.S. ambassador to South Africa.
It was 1986, and apartheid was tearing South Africa apart. The country was on the verge of what seemed like a civil war. Into this mix, President Reagan appointed Edward Perkins, the first black U.S. ambassador to South Africa. He tells his story in Mr. Ambassador: Warrior for Peace, but I'll post an excerpt here of the Juan Williams story I linked above:
Perkins was instructed to step up to [South African president P.W.] Botha, who always tried to physically position himself above the incoming ambassadors.
"The president's aides didn't know how tall I am," Perkins said. "We ended up looking up straight at each other — right into each other's eyes. I could tell that he didn't intend to avert his eyes, and I didn't intend mine either — and then when I handed him the letter of credence, he had to look down to take it — so he lost the battle that day."
Perkins told Botha that he planned to travel across South Africa to meet with as many people as possible. He recalls the conversation that followed.
"Then [Botha] stopped me and he stuck his finger in my face and he said, 'Listen. I've heard about you. I don't want you getting involved in our affairs. You understand?' And then I said 'Well, Mr. President. I'm here as a representative of the American people to the people of South Africa.' And he said, 'You didn't hear me did you,' and he stuck his finger in my face again," Perkins says. "At this time, he was kind of shaking, he was so angry. The president and I were never to have a civil conversation during my entire time there."
I recall reading in the Malcolm X Diaries that one of the formative sights that the young Malcolm saw was a delegation from an African nation, dressed in traditional attire, being allowed to dine in the whites only section of a restaurant, while Malcolm and his friends were denied entry.
He went on to write, if I recall correctly, that it was at that moment that it became clear to him that it wasn't merely the colour of his skin that led to the discrimination, it was because of the history and the stain of slavery
http://www.amazon.com/The-Diary-Malcolm-El-Hajj-El-Shabazz-ebook/dp/B00QMQ6KOG
The South African authorities created the designation of 'honorary White' to deal with this issue. Wikipedia article
While the Wiki article deals mainly with the redesignation of certain Asian nations, this was also applied to, for example, diplomats from the four notionally independent Bantustans (the TBVC states), allowing them to live in White-designated areas, have their kids attend White schools etc.
There may have been fewer problems in Rhodesia, because the Rhodesian authorities tried to maintain the pretence that they were not racist, merely that the African majority in Rhodesia was unready for political power. The official fiction was that blacks in general were not inferior per se, merely that the ones in Rhodesia were. In practice, of course, this meant deprivation and oppression for Rhodesia's black majority, which is the essence of racism.
The first part of this video shows black British comedian Charlie Williams entertaining a rapt Rhodesian audience at a nightclub in Salisbury during his tours of the country. In this case, it seems, the "British" was more important to the Rhodies than the "black"; they may have treated foreign ambassadors slightly better to try and preserve their narrative that they were anti-communist and anti-chaos, not anti-racist.
"This, as the manager will tell you, is a multi-racial club - except for Africans."