Can I download mixes to my Apple iPhone or iPad?

Yes of course,

Apple, after many years of not allowing downloading to its devices, has finally set up a download folder, which allows you to download mix links to that folder. Once you get the download link from, click it. You will see an arrow pointing down in the upward right hand corner of the iPhone or iPad. Hit the arrow to be directed to where the mix is located on your phone. In that location, you can move mix to Dropbox or Documents 6 (instructions below) app if applicable.

The free app (Documents 6 file manager) gives you better flexibility and more options. Click the link for simple instructions on how to download/save mixes on your Apple iPhone & iPad using the Documents 6 file manager.

How to download audio music directly to your iPhone or iPad without using iTunes