get the casing unscrewed off the bottom of the shifters, blast it through with 3in1 or GT85 and just go through the gears.
will be easier to do this with the shifter off the handlebars but with the cable still attached so you can provide some resistance when changing up through the gears (hold the gear cable so that your hand is doing what the rear mech would)
are these the M570 shifters?
if its been sat for 4 years the grease has probably dried out.
sounds fixable to me
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