I flush through with undiluted blue screen wash, don't use the 'flavoured' ones as they appear to contain some sort of citrus oil which I found to affect paints.
Once flushed through, I remove the back handle, sclacken off the needle nut and just back the needle out 1/4 inch. I then remove the nozzle holder at the front and then push the needle forward and extract it from the front. This prevents paint being drawn back into the rear of the brush.
The needle is then cleaned and put to one side. I get some kitchen towel and twist one corner to form a thin spike of paper which I insert into the back of the nozzle to clean it.
The nozzle and holder re fitted to the front and the needle re inserted from the back and tightened up. I then flush through again with screen cleaner. Repeat the above if you are not happy you have all the paint out but it usually works ok.
Lastly I add some thinners, normally
Tamiya, just a little, to flush through again. If I know I am going to use a specific paint next I will use the thinners appropriate to the paint I am going to use next in that last step.
If I have used lacquer or enamel paint I do the whole flushing process with cellulose thinners, be sure to be well vented if doing this. I will also use the cellulose thinners if I have had issues cleaning acrylic paint.
As I mentioned in my earlier post, when you can, invest in the H&S cleaning kit. The reamer alone is worth the price especially if you get gummed up with varnish or such.
Took me longer to type that than it does to thoroughly clean my
airbrush.