In a functional language, neither

(though new data is created as a result of running the function, technically this is guaranteed to not affect the inputs due to the function having to be pure)

(perhaps this is excessively pedantic)

let's just map pedantic->precise and call it a day :)

When running the routine, is it typically the function that changes or the input data that changes?

If it's the same function running on different data, then you are applying the function to the data. If it's the same data running in a different function, then you are applying the data to the function.