Everyone has their way of doing it, but here's what works for me.
What works for me is to transfer them when the soil is almost dry, and when the cup is almost rootbound. Using your fingers, gently massage the cup all the way around, loosening the roots from the wall of the cup. Do this until you see all of the roots unattached from the cup. Then, while supporting the trunk, turn the cup upside down and gently press around the cup until the cutting and soil slide out. Place the cutting into the prepared new pot. Works really well for me and I don't damage any roots. The advantage of having the cup filled with roots is that it will support the soil mix and prevent it from falling apart when you slide it out. Just don't do it right after watering or the mix will fall apart during transfer, due to the weight. I will try to make a video next time I do this.