It really depends. If it’s an official remix, then you can pretty much guarantee that original files were given and used.
For less official remixes, a producer can use software or other means to isolate the instruments or vocals they want. Or, depending on the popularity of the song, you can find isolated tracks/samples already online. Some artists will release instrumentals or vocals only tracks themselves as a way to encourage remixing. Jay-Z, for instance, released all the vocals of his “Black Album” specifically so that people could remix it as they so desired.
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