Start by removing the colour cup (or bottle, whichever you're using) from the brush.

Holding the brush as pictured (closeup below), use your spray bottle to run water through the brush via the inlet. Keep shooting the water until it runs clear from both the inlet and the nozzle.

Closeup - hold the brush so that the trigger is pulled back to allow water to run through the paint channel, but keep it up so that you're not running air as well -- be careful, as a mistaken press on the trigger can make quite a mess.

Do the same with the colour cup - shoot water in both the cup and the feed tube. Use the water pressure to loosen the paint from the walls of the cup. Keep moving back and forth between the cup and tube until the water runs out clear.

You might find a bit of a residual ring of paint along the sides of the cup -- grab your paint rag and use it to clean up this ring. That which doesn't yield to the rag isn't likely to come off while painting, so I tend to just leave it until the final cleaning for the day. You can decide whether you want to leave it yourself. (If for example you're going from black to white paint, you might want to be sure you've got all the black residue out of the cup first. Going from yellow to red though might not be as crucial.)

Reassemble the gun, squirt a small amount of water into the cup, and spray it into your rag or out through your shop air filter. This will usually catch any small residue of colour that might have been left behind.