I'm not sure if it answers your question, but I used to work with a guy whose father was a diplomatic courier. He used to deliver, by hand, an attaché case of documents to embassies in different parts of the world. (He flew from military airports btw.)

When he arrived at his destination he would be met at the airport by embassy staff, and taken directly to the embassy, no customs, no visas, no searches, no passport stamps, just a passport and ID check. He would also be considered to have diplomatic immunity in whatever country he was in, for the duration of his stay.

If, however he was not there on diplomatic business, he would have to have all the necessary paperwork, visas, etc., just like you or I.

I'm not sure how it would work with an actual diplomat (as opposed to just a courier), but I would imagine that it would work in a similar way.