Night shots of moving subjects is more a matter of technique than the camera. A better camera never hurts. But you need to accept the reality that moving subjects in low light are difficult to capture. You will either have lots of noise, you'll need to pan (follow your subject with the camera), or you'll get blur. There's no way around it. No matter what the manufacturers try to tell you, cameras aren't magic and shooting at night is really, really difficult.

I'm not really sure which to choose, either. I am liking this pocket superzoom category more and more, though. Between Panasonic and Fujifilm I think it's a choice between image quality (Fuji) and features (Panasonic). I don't know if it will help, but I wrote articles on both cameras:

Panasonic Lumix DMC-ZR1 Digital Camera >>
Fujifilm FinePix F70EXR - Pocket Superzoom With Super CCD EXR Sensor >>

It's a tough call. I know I've been a little frustrated with Panasonic's image quality in the past. But the features are getting so good I think I could forgive a bit of IQ compromise. My concern with the Fuji camera would be speed. I know the Panasonics are pretty quick. But I'm not so sure about the Fujis and it's been quite a while since I've used one.