These are my steps for routine cleaning after use, others may well disagree, but it's what has worked well for me these past nearly 50 years. After job is done I spray 2-3 thinner rinses through, at higher pressure of 25-30 psi. One would think there can't be anything left. There is! After spray rinse I remove the regulator and nozzle, push the needle fully forward and give it a good wipe with thinner on clean cotton pad. Plenty of thin paint residue there on close inspection of pad, prior to pulling the needle out the back.
If I am certain there are no rough machining burrs on the rear of needle, I'll actually pull the needle out of the brush by forward removal. No chance of scarring of the Teflon internal bearing that way, and I won't be pulling any residual paint on needle back through the brush body. I use the Don Wheeler adage, only hold the brush nose down when working with cleaning agents, otherwise it can affect the air valve internals.
Long tip pipettes work well for spraying a good rinse through the needle bearing, also helps to rinse the nose section of the brush a bit more as thinner flows through. I use long toothpicks sanded to the same size as the interior of the needle bearing, then after dropping thinner in bearing I lightly move the toothpick back and forth a few times, then another rinse with pipette, then it's clean. The needle would work as well, but may be exposed to damage by handling for cleaning chores.
With regulator and nozzle removed I use cotton buds and interdental brushes for thorough cleaning, tiny paint brushes will also get into tight places. A short soaking in thinner prior will soften and begin to dissolve those residues, very little scrubbing is needed then. Depending on the brush design, the nose interior can be cleaned with cotton buds, brushes or whatever gets you in there.
Some of my Badger brushes have the tiny tip that is difficult to remove/install, (155, 100G, 200,) etc. Only on those models do I not remove tips for cleaning, a good rinse and LIGHT scrub with sized toothpick will work well.
An item I consider important for final stage prep is brush lubricant. I use the Badger product but there are plenty of other choices out there, as long as it's fully compatible with all of the components it will come in contact with.
Many of my brushes are 40+ years, Paasche and Badger, all work well today. Even though a cleaning disassembly does admittedly take a few extra minutes, for the long term you can feel assured the brush will perform well every time, lasting for many years. I consider that to be more than a good return on the investment of my time.
Patrick