BTW, the original smart object is in a RGB document, and the target document is CMYK.
So what are you talking about exactly?
Several RGB Adjustment Layers can simply not be (edit) converted to CMYK.
Or do you expect the Adjustment Layers inside a SO to affect underlying Layers in the containing doucment?
That would not be possible as an instance of a SO is represented as one pixel Layer in the containing doucment.
Could you please post some screenshots or mock-ups to clarify your point?
So if you have as smart object layer with some and some linked adjustment layers that are clipped or un-clipped to smart object made into a group. You can target the group then use menu Layer>Duplicate Group (Shift+Ctrl|CMD+Q) shortcut to duplicate the group into your other document. Then in the other document you can replace the embedded smart objects layer embedded contents. Just remembered "replace contents." that replaces the embedded object it does not change the associated transform of the layer so you better make sure the replace object size is the same size of the original object was. All object are not created equal in size. If the original object is seceral layers or an image file like a RAW of jpeg file and you replace it with a PSD file the replace contents will be the PSD File snd have the size if the psd file. The Associate transform was recorded for the size of the original object. If the new content does not have the same size the smart object layer size will be different the the original smart object layer size.