Weird Focusing Issues - Panasonic ?

Below are some example images, all hand held and interestingly the slowest shutter speed by far produced the best results. Although handheld, these have recently been very repeatable tests. I take your point about razor thin DoF and you are right on that score, I have spent a few months discovering that issue and in most cases have it nailed - Many thousands of roller derby shoots taught me very quickly mainly the hard way
But the 100-400 was a whole new ball game, wow and people thought f6.3 was slow, they have no idea. Anyway here are some shots...

Click photo for bigger image: 'lens_test_g1_plus_100-400_typical'. Now I am not surprised this isn't better considering the camera and distance and possibly haze, but ...

Click photo for bigger image: 'lens_test_g1_plus_100-400_best'. This was the best of four shots from the G1 and for 1/4 mile and 12mp I think its pretty reasonable, however ...

Click photo for bigger image: 'lens_test_gx8_plus_100-400_typical '. Virtually all my GX8 long shots with the 100-400 were of this quality. But if shooting nearer objects, then they were most likely to turn out as below....

Click photo for bigger image: 'lens_test_gx8_plus_100-400_closer range '. Most handheld GX8 shots at 20 metres with the 100-400 were of this quality even though it was shot at a slow shutter speed. Because this was at f6.3 not all of the head is sharp, but most of it is and wow, it is sharp.

So I am beginning to wonder if the stabilisation could be partly responsible, in that I am beginning to think that interaction between certain focus areas and certain subjects are causing an upset.  On some scenes it is rock solid, but others it seems to struggle with. I have noticed that AFC is slow to focus and also hunts badly on a still subject without much detail. AFS however is so fast it has shocked all my Canon / Nikon friends. 
Anyway, I shall be interested in feedback and others findings.

PS. I understand your suggestion re use a tripod, and I will try it, but it is strange that I didn't have any issues like this until 2 weeks ago, even at 1/50. And at 1/2500 I wouldn't have expected any problems. But will try it in the next few days and report back.