I never soak my airbrushes in anything.
How long are you letting the airbrush set with paint in the cup?
If it is longer than 10 minutes I would suggest spraying it out and putting some water through it, if using acrylics or thinner if using enamels, to flush out that paint until you are ready to start painting again.
I would also be careful with pipe cleaners as they can leave traces of lint in the airbrush.
You are better off buying some Airbrusher's Brushes and using them.
I personally never run a pipe cleaner or brush through the body of the airbrush as it is unnecessary unless you have pulled the needle out while paint was still in the cup and the paint flowed back into the body past the needle bearing.
“Wisdom is the right use of knowledge. To know is not
to be wise. Many men know a great deal, and are all the greater fools
for it. There is no fool so great a fool as a knowing fool. But to know
how to use knowledge is to have wisdom. " Charles Spurgeon