I like to carbonate in the bright tank mostly because I don't trust all my cornys to stay sealed with no pressure. If I plan to filter, though, I will leave it pretty much uncarbonated in the bright tank and carbonate after filtering to the serving tank.
CO2 moves from the serving tank back up to the bright tank via the jumper with gray fittings on both ends. Beer flows from the bright tank to the serving tank via the jumper with black fittings on each end.
There is some valving in the middle of the gas jumper. It's not critical, but I use this tube for other things and find them convenient for this operation sometimes too.
Here is a different shot of the same gas jumper hooked up to two kegs for pressure relief fermenting. For racking, the keg pressure tester in the picture get replaced by the CO2 bottle.
If the flow slows down too much at the end (as will happen on hot days sometimes), I can close the valve nearest the serving tank and burp the serving tank relief valve to keep things moving.
Those two ball valves DO NOT have backflow preventers in them, by the way. The thing wouldn't work if they did.