Grip Shifters

[Gorgio]

Currently renovating a child's bike. It has 24 inch wheels and is a suitable size for a 7-10 year old. Anyway it has grip shifters. it is a six speed with three chain rings on the front derailleur. On the rear derailleur the shifter does not appear to move the gear cable at all when shifting from 6 to 5 and 5 to 4. I am actually holding the cable with my finger and thumb against the cable casing when testing the shift. No movement. Can not feel any resistance on the grip either when doing this. I am assuming it is just worn out and needs replacing. Any thoughts?