Right, let's try to put this to bed once and for all...
However, if the shutter speed is sync'd (no flat batteries etc), why would this not help rule out subject blur?
Because whatever shutter speed you select, the flash is faster than the shutter, the only function of the shutter is to be open when the flash fires.
So, how fast is the flash? Well, that depends on the flash and the power it's set to, but you have the Lencarta ElitePro 300 so let's deal with that. The flash duration at full power is 1/2000th second, that's a t.5 time which means that half of the power is discharged in 1/2000th of a second, the other half takes longer and makes little difference to the exposure. So what you have is a very bright, very short flash. Then the shutter closes. While the shutter is still open it will accept
some ambient light but normally it is nowhere near as bright as the flash and so makes no or very little difference to the exposure.
Oooooo and Garry, what is the max sync speed of the Elite Pro 300?
It will synch at any speed, and so will any other flash head. The limitation is your camera shutter, and any delay in the radio trigger, not the flash head.
F8 is usually selected as it's probably about the sharpest point of the lens.
Not always. Given enough power, you can shoot at F16/F22, which is what I would recommend for children. Focussing is far less critical with a wider DoF, which is very important when the little rugrats are moving about the place.
Shooting at f/16 (not F16) is just about acceptable with a full frame DSLR or 35mm film. If you use f/16 on a cropped frame DSLR then you'll get an overall loss of sharpeness caused by diffraction limitation, and the smaller the aperture, the worse the problem becomes. And f/22 is a very bad idea, unless you're shooting with something bigger, a medium format for example.
Why do manufacturers make these small apertures available when it's a bad idea to use them? Marketing I guess, because people think it's good to have lots of features. I have a car that will do 150Mph, that doesn't mean that I should drive at that speed.
My mind is also now thinking that if the ambient light is under control, flashses are working fine - is it possible to capture fast moving motion in a studio?
I'm not sure what you mean. As I've said before, a bit of movement blur in hands or hair is often a good thing because it makes the shot look more real. I remember, some years ago, when I bought a top model supercamera with 1/8000th sec shutter speed, I took some shots in the pouring rain. The high shutter speed froze the rain and made it look like icicles or bullets, it didn't look like rain at all because there was no movement blur.
If you want more blur then turn the flash power down, because lower power settings increase the duration of the flash.
If you want less blur then you're stuffed, unless you use hotshoe flashes instead, at low power, because hotshoe flashes always fire at full power but use IGBT controllers to cut the flash off early, which makes the flash duration very short. But you will have to use very high ISO, with the quality loss that this brings, if you want to use hotshoe flashes at low power.
At present you can't achieve very short flash durations with studio flash because the only one that uses IGBT controllers is the Alien Bees Einstein which isn't available in this country and which seems to have issues anyway, leaving you with a choice of hotshoe flash or hotshoe flash - or a small amount of subject blur with studio flash.
Hope this helps
Edit: Crossed with various other posts, notably Richard's
:bonk:
....:shrug:
Either way it's not usually a big deal really and if absolutely accurate colour is vital, you should always do a custom white balance or shoot Raw and correct in post. I digress... 
