I have a '13 Trek MT-60 for my son that uses a Shimano Tourney derauilleur with a 6 speed revo shifter. The problem he had was that the shifting with it was incrediably stiff, even for an adult. I figured it was the shifter and I replaced it with a 7 speed SRAM Gripshift Max that I pulled from my bike. It was a really easy shift for me so I figured it would be an improvement for him. However, the downshifting is still really stiff. Now I'm wondering if I need to look at the derailleur itself. Other than coating it in lube and hoping it works, is there anyway to adjust the spring tension on it? Would going to trigger shifters like basic Shimano EF-51 work for a young kid? He has a Islabike as well and has no issues working the SRAM grip shifter on it.
It's probably the gear cable. Over time, grease and dirt get trapped inside the plastic housing. I would recommend replacing both the inner and outer parts of the gear cable (note they are not the same as brake cables). You'll need an end cap and ferrules as well