That's the thing though. Even if you're shooting in RAW, the more you get correct in camera, the better off your photos are going to be. A custom WB would have corrected for the light right then and there, where as now you're trying to correct it after the fact but since the light was inconsistant the info that the camera got for the exposure weren't "correct" WB-wise so of course syncing in LR is going to make things worse. Syncing in LR assumes that the shots were all shot in the same light.
As part of a photography class I took a few years ago we took 2 shots in RAW, one outdoors and one indoors, using the same WB setting in camera. Then we took them into post and batch processed each one using the same WB setting...it did exactly what you described. The assumption was that since we shot in RAW and we could tweak the WB later in post we'd be able to get consistant WB simply by making the WB all the same in post. Didn't happen. It showed the importance of getting it right in camera from the get-go (even in RAW) and especially when the light is constantly changing.