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Are the handle bars on an Electra Glide supposed to feel like they are rubber mounted? When I go to push the bike by the handle bars, they move slightly upward and fall back?
Seeing you have an 02, as stated above, I would suggest putting the one piece top clamp on that came standard on 08 and later FL's. You should also check your bushings as they may need replacing. You can either replace with stock rubber or go with the poly bushings that most go with. HD and aftermarket both have the poly bushings.
56298-03 $39.95 poly riser bushings
55918-08 $39.95 one piece top clamp
Handle bars shouldnt be moving unless the bolts are coming loose or the bushing is getting worn. If you have apes, need to upgrade to poly and /or a heavier duty riser clamp.