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For example, the sentence, "The Premier of China drank vodka and beer in his car with Obama." A BBC presenter would pronounce it like:
The Premieh of Chiner drank vodker and bee'h in his ca' with Obam-er.
In other words, they add 'r's where they don't belong and eliminate them where they do.
Is there any linguistic explanation for how this evolved? Are there any formal rules regarding this?
Rhotic English is a term to describes varieties of English in which orthographic R is usually pronounced, even at the end of a syllable. In non-rhotic varieties of English - such as Southern Standard British English - orthographic R is only pronounced if followed by a vowel. It doesn't matter if there is a double R or not in the orthography:
car / ka:
car park / ka: pa:k
car alarm / ka:r əla:m
carrot / kærət
Modern SSBE also allows intrusive /r/. We sometimes use this when a word ending in a non-high vowel sound is followed by a word beginning with another vowel. The /r/ has the function of separating the two vowel sounds:
The Original Poster's example
The Premier of China drank vodka and beer in his car with Obama.
In terms of syllable-final orthographic Rs, there are two followed by vowel sounds, the ones at the ends of the words Premier and beer. These will be pronounced in SSBE. There is also one instance of a word ending in a (non-high) vowel being followed by another vowel sound. This is in the sequence vodka and. We may see speakers using an intrusive /r/ between vodka and and:
ðə premiər əv tʃaɪnə dræŋk vɒdkər ən bɪər ɪn ɪz kɑ: wɪð əbɑ:mə
Whether or not 'r' sounds that don't precede a vowel are pronounced is called 'rhoticity'. Some dialects (Most of those from England, Australia, and New Zealand for instance) are non-rhotic and only pronounce 'r' before a vowel. Dialects from Scotland, Ireland, and North America are mostly rhotic and pronounce 'r' whether or not it precedes a vowel. Non-rhotic speakers do change the preceding vowel, usually by lengthening it or combining it with schwa (an 'eh' sound) to form a dipthong.
The 'followed by a vowel' rule can include the next word starting with a vowel: this is called a 'linking r'. Less frequently there is also an 'intrusive r' which is added between vowels that would otherwise blur together. When intrusive r is not used, a glottal stop is used (as in most North American dialects). Whether an intrusive r or glottal stop is used, it's generally not noticed by the speaker or by any speakers who are used to it.
In some British dialects 'a' can also be pronounced in a way that sounds to those unfamiliar with it somewhat like 'ar' when it's at the end of a syllable. There isn't really an 'r', just a vowel you're not used to.
In your example, "premier", "beer", and "car" all get affected by non-rhoticity, "China" and "Obama" are probably examples of unfamiliar vowels that you are mishearing as ending with 'r', and "vodka and" is an example of an intrusive r.
There are many regional differences in pronunciation, in particular with "r", for speakers of English (as well many as other languages), that there really is no simple answer, other than perhaps Rhoticity :
"Rhoticity in English refers to the situations in which English speakers pronounce the historical rhotic consonant /r/, and is one of the most prominent distinctions by which English varieties can be classified."
