Hi, even if I don't really know where it leads, it seems that we can move 
Objects with the depsgraph in bge.

I don't really understand for now how to combine logic and blender depsgraph, 
but this is just an experiment to use depsgraph in bge.

The code is here:

https://github.com/youle31/EEVEEinUPBGE/tree/experimental5

The performances are not good for now. and I just tested on a cube.

The pipe is weird but as it is an experiment, we don't really care:

1) The logic updates "SGNodes" matrix (KX_KetsjiEngine::NextFrame)
2) I update Object (KX_GameObject's Object) with the resulted matrix 
(copy_m4_m4(ob->obmat, sgnodeObmat))
3) I tag the Object for update with DEG_id_tag_update(&ob->id, NC_OBJECT | 
ND_TRANSFORM);
4) I update the depsgraph with BKE_scene_graph_update_tagged(eval_ctx, 
depsgraph, bmain, scene, view_layer);
5) Then I run a complete render loop (create a viewport offscreen, set DST 
variables, fill eevee's cache, draw eevee's scene, free viewport and cache...)

On my level, this is just for experimental purposes, but I hope it could be 
useful to more experienced coders for interactive mode.

After step 1), the code if the following:

http://pasteall.org/878033/c (Sorry about the dirty state of the code)

youle
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