Flying-free - line a plastic bowl with a towel, fill it with room-temperature water containing some bio-violent (enzymatic) washing liquid and sit the vase in it for a few hours.
Your dirt will lift off - a delicate make-up brush, such as an eyeliner one, will allow you to clean inside the crackles once it's all loosened by the enzymes. You can isolate a single hair from the brush to poke into difficult bits.
I managed to get the base of one of Richardson's uranuim plant things spotless - including all the tiny bits in the convolutions.