Morphs can be saved as morphs.
"Save As/Support Asset/Morph Asset"
If you save a morph for a custom character, then I think is only going to be available for that character. If you save a morph as part of a scene, then it's probably only available for that scene.
Most morph creators create and save morphs for the base DAZ character (G3F/M, G8F/M etc). That way, all characters/actors based on that base character inherit the morphs for that character as well.
At least, that's the way I understand it to work.