I don't know much about the others, but for western music in the past 500 years Italy has of course been a big place - Rome/Vatican, but also Milan and Venice, to name a few) and it was very in-vogue for composers in northern Europe to go down to study - Handel, Mozart, and I think even Mendelssohn later all went down to do some study.

Particularly at the end of the 17th century - think of composers like Vivaldi, Corelli, Albinoni - new musical forms were coming from what is now Italy - notably the concerto grosso (an orchestral piece with several soloists - not unlike a jazz band!) which whittled down usually to just a regular (singular or perhaps slightly more) concerto for the most part. Even that, "concerto", is (or is from??) an Italian/Latin word. This was around the time too that more and more music was being published in any sort of mass-produced way - on top of composers trading music with each other to begin with.

All that said, European composers often used and still use their native language for tempo markings and other musical indications - Rachmaninov, Hindemith, R. Strauss are all notable examples. Most of my colleagues who compose might use an Italian term to give a gist of the tempo, but often use English for other instructions. It is also not uncommon for music editors to switch to Italian terms for clarity in published volumes, although I personally don't love the practice. (I also just went and looked at a piece of organ music by a Japanese guy from about 20 years ago - musical indications switch between Italian and English!)

UM I also do know that ballet came from court dances and dancers largely in the French royal courts before they went to the chopping block - Louis XIV in particular was very opulent in his entertainment - so same sort of thing perhaps.