I use a Badger 150 double action siphon feed and acrylics, so your results may be different.
I use Windex (actually a generic ammonia based window cleaner, MUCH cheaper than Windex) to flush through the AB into a waste container, then follow up with a shot of distilled water, also into the waste container. I re-use windshield washer containers for my waste.
Since my AB is a siphon feed, I can use different bottles or paint cups for each colour. I pre-clean the cups while it's still attached by pouring cleaner into them and squirting that into my waste container, and do a full clean and dry routine when I'm finished. If I'm going to be doing multiple colours, I fill a old sour cream container or bowl with water with a bit of cleaner mixed in for temporary storage until I'm finished. That keeps the paint from drying out in the siphon tube. I use moistened cotton swabs to wipe out the connection to get out any remaining paint and/or cleaner, and also to clean up the nozzle cup (retracting the needle first!).
I don't use filtered water, only distilled, I live in a hard water area and am concerned about mineral buildup inside the AB. If you live in a soft water area, or have a water softener you should be fine. If you notice a whitish buildup on your taps and tubs, use distilled! Brita's are good for bacteria, but don't catch the calcium! Distilled water is relatively cheap, and fairly easy to make yourself (boil a large container of water, let cool for two days without disturbing, then siphon, never pour!, all but the bottom 1-2". This removes most sediments). Since a gallon jug lasts me about a month, and costs less than $2, I just buy the stuff! I can then spray the next color. I use the exact same method to store, and add removing the needle and wiping it down with the cleaner, then smearing with Medea Super Lube (or your favorite needle lube!), before re-assembly for storage.
I never put thinned paint back into the original paint container. I always thin my paints, and am starting to figure out the coverage required so that there is little or no waste. I use a leftover paint bottle to store bulk thinned or mixed paint in, and the little bit that is left over gets squirted into the waste container.