You can transfer the morphs to varying degrees using Generation X, a plugin for Daz Studio which allows you to take old morphs and convert them for use with Genesis. It’s a bit of a mixed result, as Genesis has a lower poly count, but they tend to work reasonably well.
Clothing items can be autofitted to Genesis and are converted when you fit it to the character. Skirts and boots tend to be a bit off though, so it’s not a perfect conversion. In some cases I found myself exporting the mesh as an OBJ and then re-importing it using the transfer utility to retain the shape. Hair tends to work fine, but might need some minor tweaks, and prop hair will need to be parented to Genesis’ head to follow it properly.
Other than that, yes, many things are indeed forwards compatible. It’s only backwards where things won’t generally work.