I want to know how these schools spread to & across the West, any interesting circumstances?
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Not a complete answer (and limited to the US, and speculative at that) but ...
re Mahayana: Likely the first introduction of Mahayana to the US came from the importation of Chinese labor used for the construction of the railroads (c 1865).
@ 1938 Dwight Gottard's A Buddhist Bible was instrumental in popularizing a mix of Theravada, Mahayana/Zen Buddhism for a time. (The popularity of Buddhism in the U.S. seems to come and go.)
Later the Beatnicks re-introduced Zen from Japan after having come into contact with it while enjoying R&R during the Korean war. This was much popularized by Jack Kerouac's reading Gottard's "Bible".
As for Vajrayana and Triratna I'm not sure these can rightfully be said to have spread in the west as of yet.