When I pull the shifter for my front dérailleur the cable doesn't react. Even with tension on the cable.
Now I made the cable loose. Also my barrel adjuster is disconnected, maybe the problem is related to that.
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If the housing is loose and falling out of the barrel adjuster, either the end of the cable as come loose, or the housing has come out of a frame stop (or the shifter itself, seeing as you have the bar tape off), or the cable has broken (or cable end in the shifter may have detached). There is also a chance that the shifter is broken.
If you've grabbed the cable (not the housing) and worked the shifter to check the cable is being pulled by the shifter, and it's not, the cable probably is broken or the end detached. Try pulling the cable and see if it comes loose.
You can also check the cable has not come out of the derailleur pinch bolt (or pulled through it). Also, follow the cable path from the shifter to the derailleur and make sure the housing is properly located in all stops.