If you aren't comfortable taking it apart, what you could do is use the straw and spray little amounts of it inside many times, doing a few moves to distribute it and blow some air in between to make sure it doesn't stick together.
I agree with ejisfun that that it could leave the white residue if lubricated. What I did was twisted the thinnest layer a little and stuck the straw into the middle (and made sure it was deep enough) and squirt some in there, distribute it while blowing out the fumes, and repeat it until I was satisfied with the smoothness. Make sure not to squirt too much at one time so it doesn't come out anywhere else!
There is white on the sides of my cubies, but that's more scratches than lubricant film
Be warned though, the smaller layers would be harder to turn than the larger ones, giving it a feeling of uneven tension, or something like that
I'd rather put up with that and have a smooth puzzle!
Edit: Oh, and if you put too much lubricant in the puzzle like I did, it might get a little uncontrollable!