I have found that it is possible to get the motions back into DAZ studio via BVH, but you're going to have to do some work with it first in something like Motion Builder or Ikinema WebAnimate. (Wendy's right though --DAZ FBX Import is terrible and completely worthless for getting anything other than the geometry and maps.) I imagine that the HumanIK in iClone is not fully utilized when it comes to the Genesis 3 Figure. What I find works for me is Characterizing a Genesis 3 Male and Female in
Motionbuilder, and then bringing in the Motions and getting them to work there and then bringing them back into DAZ studio From
Motionbuilder as a completely characterized BVH. Webanimate has a similar process that is as good at remapping BVH to different skeletons.
Motionbuilder's expensive --Webanimate is only $99 dollars.
There is a third option, and that's creating a Genesis 2 replica of the figure in question --or as close to the figure general size and proportions as you can get it, importing the motions from iClone on the Genesis 2 skeleton and then converting it inside of DAZ studio using a
DraagonStorm's Script at DAZ3D or this
ShareCG Script. They aren't perfect conversions(They get you about 90-95% there but you will have a few adjustments to make to get a finished result), but with a little work you can iron out the kinks and minimize the headaches..
Flipbook Market creates a lot of animations for DAZ studio and posted a tutorial on how to use Genesis 3 with
Motionbuilder you can view
Here It's a bit lengthy, but there's some good information on doing it.
Webanimate offers some information on doing it as well, but to be honest I don't use Webanimate. I've heard a lot of good things about them.
From what I understand, this isn't uncommon with BVH files and if the bones are not a one to one match, there are problems. I would like to point out one thing that I see in iClone that may be part of the issue and that's NOT including the twist bones of the Genesis 3/Genesis 8 figures in the HumanIK mapping --even though HumanIK does allow for twist bones in Characterization. In the templates from iClone, they are not used. You can see in this diagram where the twist bones options are located --both in
Motionbuilder and iClone 3DXchange.

I haven't tested this, but I have a feeling if they were to be included in the characterization, you might be able to export motions from iClone to DAZ directly without unexpected results. It would more than likely require that you forgo the automatic character setup and map the bones manually, saving your own template if it works OR you might let the characterization go through and simply connect them once it's done and then save it --but I'm not sure that will work. With a little experimenting, this might be a workable solution. I simply do my work in
Motionbuilder because I have it, and it gives me a LOT more control to do it that way, and I'm not really keen on DAZ studio's lack of growth when it comes to anything besides figure geometry. There isn't a foolproof 100% way to do it error free, but you can get it if you apply a little problem solving. Once the motion's there in DAZ studio, you can export MDD or Alembic, capturing all the detail and the Joint Control Morphs like you were talking about.