We can embed details in families. Take a structural door opening that shows the details for the lintel over. It can be inserted into any wall, and we would like to adjust the detail based on the thickness of the wall. Secondly, we also want the detail to be switched out with a family type parameter if necessary. Is this even possible?
I create a detail family that is parameter driven and then see if we can adjust it based on the width of a wall. Setting up the family with a shared instance parameter and then I set the detail item family to share.

When I try to load the family into the wall family, it refuses to share. Hence, you cannot drive the parameters of the detail from within the host family. So, it is better to create the details directly in a section in the Door family.

Now, working under the assumption that we would like to switch out family typed in the door, we can create family types around the typical wall thicknesses, but we would have to assign the family type parameter as instance and adjust as per wall thickness. It cannot be driven by a reporting parameter that references the thickness of the wall.

In conclusion, you can use family type parameters to reference detail item family types, but you cannot share them in or control their sizes using parameters.
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