you'd expect them to have brought the nagaswaram/zurna/mizmar/bombarde/shawm with them
Maybe they did, but as the dance troupes over the years proliferated they became populated with indigenous performers who would have used instruments they could make and play.
And would these Asian itinerant acrobats settle in the British Isles or would they keep following the money? Tick both of the above but not in every case. Fashions is fashions. And they morph. Affluent audiences get tired of the "same" and warm to the novel. Things change. Pick yer year. But the evidence shown to me was that it all moved down market morphing as it went. And proliferating as any thing (cf technology) does when it becomes affordable.

Is anyone here seduced by the word "designer"? Thought not. It has lost its original reason as it moved down market.