Originally Posted by
Artmo
I have this problem each year on a bike I keep in the UK. The bike is not used for about 12 months, but I simply spray WD-40 into the lever mechanism and work it to and fro and this fixes the problem.
This is also what I recommend. But you probably do need new cables and cable housings. The housings collect dirt and corrosion and should be changed with new cables.
When shifting to smaller cogs the derailleur spring is pulling cable from the shifter. Any excess friction along the path of the cables can cause shift hesitation.
But try the WD40 trick first. If it helps you know you have excess friction in the shifter.