I disassemble the whole filter rig and soak it in Starsan. While that is soaking I sanitize the destination keg and purge it with CO2. My filter setup has MFL fittings for the QDs, so after re-assembling the filter and housing, I connect one QD to the purged keg beer out side, and loosen the other QD so I can purge the filter with CO2 using the purged keg beer out. After purging for a bit, I tighten the fitting up, and the rig is ready to go (the purging process also helps to get excess Starsan out of the filter).
I relieve the pressure on both kegs (don't forget to do this!), connect QDs of the filter to both kegs. Then I twist the relief valve on the destination keg so it will stay open. If the valve won't stay open you can unscrew it and take it out. Then I connect my regulator to the source keg set at 0 and turn up the pressure JUST until the beer starts to flow, usually about 2 PSI. It takes about 15 minutes to push all the beer through to the destination keg. Once finished I put the relief valve back on the destination keg, purge just in case there is any O2 in the headspace, and it's good to go.
I don't recommend filtering carbonated beer, as it will foam up a lot. If you use a 5 micron or larger filter, you will get some yeast out along with the proteins that cause chill haze but will leave enough yeast to carb naturally. If you want to sterile filter (1 micron or less) and carb naturally just add some dry yeast to the keg. Any yeast will do, you only need less than a teaspoon to add along with your priming sugar so a single pack of ale yeast will last you awhile if you keep it sealed in the refrigerator.