AutoFit has the morphs there and just keeps them: they are stored as deltas, differences from the base shape, so in many cases they will still work reasonably well on the modified shape. When you have an OBJ it doesn't have the morphs, so they cannot be transferrred as deltas; exporting the original morphed shape and simply importing that as a new morph will bring the literal shape across, rather than the differences, and so will push the item back to its original shape plus the morph rather than its new shape plus the morph.

It might be possible to transfer the morph deltas by file hacking (copying and pasting between the two .dsfs, or perhaps better creating a new empty morph from the right-click menu while the Parameters pane is in Edit mode, saving as a morph asset, and then copying the deltas across) but it's not something I have explored.