I'm going to assume you are using Zbrush. I've done quite a bit of morphing of fibermesh hair conformers, setting the morphs up to superconform or follow the morph dials of the figure. Most of my work there was with brows and eyelash conformers.

If you are trying to use some copy morphs process I would forget that route. You'll have to set up the morphs by creating the matching morph(s) by hand in your sculpting app and naming them identical (internal name) as the morph(s) contained in the target figure.

You'll want to mask or freeze the underlying mesh so that only the fibermesh and cap are effected by your brush. The introduction of a cap actually makes the process of creating fibermesh morphs a bit more tricky. You'll want to have your brush set up so it effects volumes rather than surfaces.

Zbrush's brushes by default effect volumes...

...so the brush is grabbing both the fibermesh en masse and the underlaying geometry. There's no cap in these examples so the brush is effecting the head mesh too. Masking would probably be best with Zbrush.

Blender's sculpting tools are very similar in action by default. That is by volume. There is some automasking by topology options but just straight selection or brush derived masking is best here...

Mudbox is a bit different in that it's brushes effect surface/volume or just surface or just volume. It's my preferred app for morphing and painting/projection for Poser for a variety of reasons. Here the brush is effecting volume. Masking or freezing would be the approach I would go also...

3DCoat is a bit of an outlier for me. At present I use it (or am learning to use it) for retopo and painting and UVs. By default it also effects volumes...

Probably not what you want to hear but if I were developing a fibermesh hair product with the intent of distribution I would have skipped the cap entirely. Downside is that a manageable fibermesh hair is a bit sparse and you could fill in that sparseness with a textured cap. You could skip the cap and have the "fill" hair texture as an option to the included L'Homme head textures. Morphing both the finicky fibermesh AND the cap to match numerous morphs seems like a ponderous additional thing to labor at. Just my opinion.