I attended an all-Mozart program the other day and I was wondering if all the pieces he wrote are performed by exactly the same number and kind of instruments you would hear today. The French horn player said the piece he played was originally written for violin, and was arranged for horn by his instructor. Is this done often?
Arranging and transcribing is not uncommon, for reasons such as, for example:
Sometimes (although not, I think, not in the case of Mozart), we are not sure what instrument the piece was written for, so different instruments that seem to suit the piece well may be used. A well known example of this case is Bach's The Art of the Fugue, for which there has been performances by many different types of ensembles, solo organ, etc.
Even the composers themselves adapted a certain previously written piece to a different instrument, to experiment, to quickly fulfill a commission, etc. An example I recall from Mozart is the famous Clarinet Concerto, of which at least the 2nd movement had been previously written for flute. Apparenty Mozart had written the piece for flute as a comission, but he was not very fond of the flute as an instrument and reappropriated the theme for the clarinet concerto, with the magnificent result we now know.
Certain instruments lack a rich solo repertoire, and pedagogues or great performers of the instrument are always looking for pieces written for other instruments that can be easily adapted. Andres Segovia, for example, as part of the work he developed to launch is career and divulge the classical guitar, adapted many Bach's pieces to guitar, of which the Chaconne became famous.
There are a number of ensembles which perform regularly on period instruments. The same compositions they play, whether arranged for modern instruments or for different ensembles, are adapted depending on the current need or the arranger's own interests.
For that matter, lots of music is "borrowed" (violas, a clarinetist is looking at you) to be played by different instruments. I happen to know a fellow who played the Bach Double Concerto as a tuba duet.
And BTW, over at imslp you will find multiple arrangements of many chamber pieces for different instrument groups.