Since I've been through a lot of oiling with the ITG air filter I use on my Z, I'll chime in.
No Toil makes a great foam filter oil - you spray it on the dry (clean) filter generously, rub it into the foam carefully, then squeeze out the excess. The foam will be very sticky, and can make a little bit of a mess on anything it touches while installing, but that stickiness is what catches the dust and keeps the inside of your engine clean. The No Toil oil is nightmarishly difficult to wash off without a proper cleaner though, which they sell. Interesting stuff that cleaner, it's a powder that has a lovely burning sensation when you use it to wash the oil off your hands after forgetting to wear gloves when oiling...
ITG has actually recently switched from recommending No Toil and now recommends "Rock Oil Factory Eco Foam". This oil is actually water soluble, meaning you can wash it using water and a touch of regular detergent. No need for a fancy foam filter cleaner. I'll be switching to it. Coast Fabrication in California sells it, good people.