The basic idea is to filter the paint - this way you only get good liquid paint and no clumps in your airbrush pot. The filter medium that I ended up using is cheesecloth. I just doubled over a sheet and did the following (the photos below are from a dry run, the process is a bit messy and I didn't want to try and take pictures with wet paint all over my hands) -
Make sure your pot is nice and clean.
Double over the cheese cloth and push it down in the pot.
Use whatever liquid you are going to use to dilute the paint in the next step. In this case I was using a 1:10 ratio of Flowaid and distilled water.
Wet the area of the cheesecloth that is inside the pot.
Pour in your paint. Don't worry about the clumps.
Slowly start to lift the cheese cloth out.
Once you get the cheesecloth above the rim of the pot, gently squeeze it to remove any excess paint left in the cloth.
At this point you will have a nice clump free paint base in your pot ready to add the rest of your thinning liquid to.
Easy as that. This process took about two or three minutes and worked like a charm. I didn't run into a single clog while using this batch of paint - and it was even the bottom of the barrel.