Welcome to the forum!
I'm not sure if I understand what "mushing" is. But it sounds like you want to make a change to one of 3 objects, and have the other 2 objects change automatically. Correct?
Inkscape doesn't have any functionality exactly like that. But you can get pretty close with clones. So if you have a 3 piece character and clone it, you can change the parent object, and all the clones will automatically change.
Or, if the objects are like the example you showed, where all the pieces of the character share all or most of the same nodes, but consist of a fill object, stroke object, and a modified fill object, then you should be able to achieve essentially what you want by selecting the same node for each object.
So each time you want to make an edit, select all 3 objects, switch to Node tool, and drag a selection box around the 3 stacked noded.