Try setting shutter speed to 1/50 - 1/60th sec for a 24 frame per second GF-1 and 1/60th - 1/100th for 30 FPS on the GX-1.
Shoot in manual exposure mode not in P or program or IA. It may help.
These cameras aren't so advanced in video mode. The GF1 doesn't shoot at 24 FPS neither the PAL version of the GX1 shoots at 30fps.
I know I can shoot at 30fps (NTSC recording) with the GX1 if I use the hacked firmware, but it was giving me trouble playing in-camera MP4 files.
I tried to shoot in P and iA mode, but the result is the same.