I realise that I'm starting to sound like a broken record, but if you want a good cleansing oil you don't have to go K-beauty. I'm using the Body Shop Camomile Silky Cleansing Oil thanks to a Caroline Hirons recommendation and I truly love it. So much that I'm not likely to even try another oil for a good long while!
Even the notoriously bitchy stringent Paula's Choice Beautypedia has nothing negative to say about it. They try, of course, and mark it only 3 stars out of 4, but the negatives are tenuous to say the least. It's the worst thing about that brand IMO, they are the Demontors of the skincare industry. Shame as their BHA lotion is very good.
TBS cleansing oil has a light herbal/floral scent which I really like, and obviously feels very 'oily' to the touch, but nicely IYKWIM. I use about 1-2 pumps. It dissolves makeup in seconds, and is very gentle around the eyes. I then wet my hands and emulsify the oil, so it becomes a little creamy. Wash off with lukewarm water. It doesn't require a washcloth if you then use a foaming cleanser - I've been using another TBS product, the Aloe foaming wash, or Su:m37 rose stick.
None of these products are 'stripping' and all leave my extremely sensitive skin feeling clean and soft. I actually rate TBS oil above the Emma Hardie Moringa balm which I also use, and was previously very happy with. Which is great because TBS oil is £10 for 200ml which lasts ages, and the EH balm is nearly £40 for 100ml. So £ vs £££.