I'm using strand based hair to make fur clothing on my Neolithic outfits. I use dForce to drape the clothing, and when strand based hair is created on the clothing surface it will move with the dForce enabled surface, but the hair itself will not move. There is a way to apply dForce to the strand based hair itself, but Daz only makes that functionality available to PAs who sell in the Daz marketplace.

As far as the hair moving when the object moves, the trick is to NOT parent it to figure surface, just create it it on the figure surface. In the case of a clothing item, if you load the clothing item with SBHair into the scene, then fit it to the target figure, then it will move with the figure and stay attached. If you do the usual click-to-fit on the clothing, then that breaks the association between the surface and the hair. It's weird, but maybe Daz will fix it in a future release.

Hope this helps.