I keep two bottles of "thinner" on my bench- mineral spirits for when I use enamel, and isopropyl for when I use acrylics. I keep a homemade cleaning chamber on the bench. This is a half pint jar with a plastic lid on it. I drilled a hole in the lid for a short piece of plastic plumbing pipe (half inch?), glued to the cap. On top of the length of pipe is a 45 degree fitting, so I do not need to aim the nozzle straight down. I spray the thinner into the nozzle and the jar collects the old thinner.
I keep a box of kleenex handy to daub the bottom of the pickup tube (I use a suction feed airbrush). I can see by the color of the daubs whether the pickup tube is clean of paint.
I do not run anything through the passages of the brush. If I disassemble and clean the only thing that touches the insides of airbrush parts is to occasionally wipe the needle with the kleenex, or to wipe out the front of the nozzle with a q-tip soaked in thinner.
When changing paint colors I only run thinner through the brush and backflush. Only dis-assemble when done for the day or with that model.
You will use a lot of paper toweling, Kleenex, and q-tips, so keep a good supply on hand.