A warm welcome to you, great to have you here.
If you could let me know if I have basically the network as you described it.
I cached to alembic a cube transforming up 10 units on the y axis.
In a new scene I imported the alembic and named it pCube_DUP.
From there I have four more duplicate cubes.
All of which are connected inMesh -> outMesh (on their shape nodes), in the above order.
I selected pCube_DEFORM (transform) and added a tissue. So the pCube_DEFORMOrigShape preserves the inMesh connection from pCube_IN, and the pCube_DEFORMShape takes the inMesh connection from the zEmbedder.
If I play this scene back, the DEFORM cube drops in space, and the IN cube goes up with the cache.
If I turn off the solver, and set the zEmbedder envelope to 0, the DEFORM geo follows the cache again.
If, instead of making it a tissue, I make it a bone, it follows with the cache.
Let me know if I have something backwards. Very keen to help your project succeed.
Typically we don't recommend having anything running into muscle geometry, but understand production needs can vary.