There are a couple of ways this might be achieved.
First, you can copy the motion from one layer of tweens to another layer with a fresh symbol. Highlight the keys, right click and select Copy Motion, then select the new symbol, right click and select Paste Motion. This applies the motion relative to the selected symbol's starting point. This option works best when the parts are set up ahead of time with identical pivot points
There is also a Paste Motion Special... option which allows you to paste certain motion properties but not others (for example, rotation could be included, but not position). You may need to do certain adjustments manually if the new character has different body part proportions. You can also adjust the proportions as needed by using Edit Multiple Frames and the Transform tool.
Secondly, you can select Duplicate Symbol from the context menu to copy the original character, then Duplicate Symbol/Swap Symbol for each individual part and add the new character art. Use Edit Multiple Frames to select a timeline of animation and swap out the symbols for each keyframe. The duplicate character symbol will keep the original animation with the new character art.